1. The Americas are going to be colonized by the English. All gold and silver from the Americas is going to be theirs.
Yes, but in some ways, that will be the least of it. Tudor England-Wales' tech lead in navigation, weapons and everything will be insurmountable. It will assume wealth is there to be had in the Americas, and definitely make the voyage. When they find the Caribbean, it will definitely be poorer and less golden and more primitive than they expect, as the Americas will be overall. But they will press on founding colonial stations in the Bahamas, Cuban, Hispaniola, and press to Mexico, anticipating gold and silver.
There they will finally find a civilization with some art, architecture and monuments, the Olmecs. And it might be the first time they encounter people even with agriculture. In any landfalls in North America - Florida for example, or on the Caribbean, the indigenous people might not yet have it.
Here are some maps of 400 BC, close enough, that include the Americas and their political situation.
upload.wikimedia.org/wikipedia/commons/0/0c/World_in_400_BCE.pngmapsontheweb.zoom-maps.com/image/631773672656519168The Olmecs at least will be wearing gold, the English will be convinced it's there, and will force natives to dig for it.
So yeah, the English-Welsh will be the European colonizers of the Americas, starting with the Caribbean, quickly moving on to Mexico and Peru, not having to worry about competition. Using their own people and the indigenous as labor forces, and when and where those are insufficient, using downtime pagan Europeans as indentures or outright slaves....which would probably not be tolerated post baptism or conversion, or downtime Africans, although all those African trade circuits would have to be re-set, so European laborers will be much cheaper.
Whenever Englishmen do get around to planting colonies on the Atlantic coast of North America, the local people will be more hunter-gatherer than agricultural, and so may be less helpful in times of food shortage.
2. The English will surely expand into Ireland, Scotland and Gaulish territory in France - but would they go much further?
Security needs, convenience and simple greed will motivate Anglo-Welsh expansion throughout the remainder of the downtime British Isles, and the subjugation and proselytization of the downtime people living there, who will be vastly outnumbered by the Anglo-Welsh to begin with.
From an English perspective, the "nice" thing is their military tech makes them non-competitive to English arms, and their heathenism means within a generation they'll be mostly open to conversion to Anglicanism because, the Irish, for example, have never been converted or committed to the "wrong", ie Catholic, form of Christianity. Nor are Jesuits, Spaniards, or Frenchmen inciting.
At the same time, they are comparative wild barbarians, who will pretty quickly resort to hit and run tactics, and neat to be conquered one village and tribe at a time, with the average man and woman probably healthier, taller, and with better individual fighting skills than the average English yeoman craftsman recruit to arms.
Frankly, I think the English expand to fill out the British Isles, and to take over Gaul - why not revive or retrofit their French crown claim, though it be meaningless to everyone else in this world.
But why should the English stop themselves from expanding *anywhere* within northwest Europe where the climate, soil, growing conditions, seasons, are comparable to England? So, in Gaul, expansion should go down to the Pyrenees and Mediterranean. But the English should be equally interested in Rhineland and the Low Countries and the North Sea region, basically all of Germany down to the Alps and at least as far east as the Oder or Vistula, and including peninsular and island Denmark.
The main problem the English will end up with from expansion, because peasants/yeoman, poor townspeople, noblemen, they'll all want land at the fringes and take what they can get, will be keeping their social order and class structure intact, and not spreading out so thin as to have labor scarcity. That's where they will want to keep downtime people around, and working, and in a lower social and servile status, even if not enslaved and brought to the light of Christ. With more land, and conquered or less "civilized" people to rule over, the lower orders of Elizabethan English society will be getting a relative "promotion", but to keep social distinctions, gifts, lands, privileges will have to be extended to an even greater degree to all the higher classes above them.
The English can assert and make claims to rule and control as far as is militarily and demographically feasible onto the North European plain. And then they can pick whatever justifications they would like after the fact.
Among the neat and tidy justifications they could use to side boundaries, even very wide ones, for their continental claims, could be claiming the right, based on their British/Welsh heritage, to all the lands in Europe inhabited by people Celtic speech - to deliver them the gospel, rid them of the druids, and uplift them. Around this time, that covers an enormous amount of Europe from Gaul, northern and western Spain, all the way east to the Carpathians and where the Danube flows into the Black Sea. See this map:
visual-search/?x=16&y=16&w=532&h=356&surfaceType=flashlight
Based on their Anglo-Saxon heritage, they can make a claim to all of Europe where Germanic or port-Germanic tongues are spoken for similar purposes. See this map:
en.wikipedia.org/wiki/Early_Germanic_culture#/media/File:Germanic_tribes_(750BC-1AD).png. Germanic languages were covering about the yellow or orange areas by around 350 BC.
3. In a few years there are going to be English trade outposts across all of Europe - as well as in Africa and Asia
Certainly. The English would also set up the round Africa trading posts to reach Asia, the Persian Gulf, India, China's warring states, and the spice islands. Hopefully all known spices will be cultivated by this point. You know what almost is not cultivated at this point? Coffee. And coffee and coffee houses might have become a thing in England already, or were soon to be one. Well, not in this world, the English will have to skip right to their tea habit.
4. What about the Greeks? The intellectual elites of the era used to hold the Greeks on very high regard but on the other hand they're still pagans.
The Elizabethan English will be interested in the Greeks, be predisposed to admire them, seek them out, and assume they have relevant wisdom they can benefit from, despite being pagans. The Renaissance-era appreciation for the classics and even pre-Christian Greco-Roman culture, extending knowledge not only its art, government, history, and secular philosophy, but even literary familiarity with the Greco-Roman pantheon....that educated Europeans, including English people, did not see as scandalous idolatry or blasphemy genuinely threatening to their Christian belief or identity, will be able to "get over" the ancient Greeks paganism without turning into raving mad conquerors, oppressors, forceful prosyletyzers and idol-smashers.
Uptime English clergy and lay people now will explain Christianity to downtime people, including ancient Greeks, and when engaged in spiritual heart--to-hearts stand by their faith and seek to persuade the ancients of their truth, but with the most admired, "civilized" people off the past, like the Greeks, English Christians will tend to use more of the soft sell for Christianity, exercising some diplomacy, tact and respect. They will apply some of this respectfulness to some of the pagan peoples well known from the classics and the Bible, like the Egyptians, Carthaginians, Romans, the Persians, and the subject peoples of the Persian Empire, whom they've read of.
The English will be far less patient and respectful of the heathenness and paganisms of Northern Europeans and downtime people on their own island, the Goidelic and continental peoples and the Germanic peoples. The English will see those paganisms and their cultures as less refined and accomplished, and especially for the Germanics of Germany and Scandinavia, a security threat leading to veneration of war and bloodthirstiness that needs to be tamed.
Likewise, in the Americas, the English will have no patience for human sacrifice traditions, and be more in a vigorous and intolerant proselytizing mood towards Amerindians up and down the line, having seen the Spanish example, and having nothing so fascinating to admire or study there like the ancient/classical Greeks and Romans.
One thing that the English explorers and merchants and later official delegations and ambassadors to Greece and Rome especially will find pretty jarring will be that when they get there, they will experience the landscape, their people, their clothing, the architecture, monuments, and statuary in vivid color, not all in just the white marble that survived to their own day. The Latin and Greek will also not sound like Church or scholarly Latin or Greek.
Are we going to see any sort of aid to Macedon? Would the death of Philip II be butterflied away?
On the one hand, the uptime English will have an admiration for Philip II, for Alexander and his destiny, and for the Greeks and Macedonians alike, and an admiration for them.
On the other hand, they will not like some of the oppression and destruction that both Philip and Alexander are visiting on some of the Greek cities.
....and they will not really have anything *against* the Persians, who, all things considered, will seem not that bad as ancient empires go. Sure, they were an archenemy of the Greeks and Macedonians, but while they last, they are a large, coherent, useful trading partner, not nearly as oppressive or cruel as the preceding Assyrians or Chaldean-Babylonians. And, when looking at the Persians from the Biblical Old Testament sense instead of the Greek lense, the view of them is almost entirely positive, with the exception of Haman, since they abetted the regathering of the Hebrews and rebuilding of the temple.
although their numbers are going to be very few and the more primitive settlements and populations in N America would reduce the level of support 
it might be worth thinking how does all of this influence Greece and Rome - this is just 4 years before the OTL death of Philip II. Rome is also already there and a big player on the Italian Peninsula.