Where to start..?
1. Optimise the game engine to run faster
2. Add at least another 5 Civs, and increase max Civs from 22 to 'all of them'
Extra civs are always welcome, but definitely optional in an expansion that has a lot more serious issues to attend to. There's already plenty of variety, and although fewer than in Civ IV, there are more factions in the base game alone than in most games out there, and DLC has added 50% more. I'd rather have Civ V's fewer but better-drawn civs (with their leader portraits accompanied by spoken-language phrases, individualistic greetings, and unique civ-specific abilities) than Civ IV's excessive number of characterless mix-n-match 'aggressive/expansionist, expansionist/creative, creative/aggressive etc.' factions.
3. Improve diplomacy. Even put back in all the Civ 4 code and I'll be happy
I doubt anyone would object to the first sentence. However, I've played Civ V games that demonstrate the existing system can be more engaging if the AI was programmed to use it properly - denouncements and declarations of friendship with relevant effects, wars that result from specific conflicts (we want the same territory/resource/city-states/victory conditions) rather than just from another civilization disliking you intensely, and that needn't result in permanent penalties to relations. Major governments in the real world, especially at advanced levels of technology, don't invest the huge resources needed to go to war just to spite someone they dislike.
4. Bring back Vassals in place of puppet states. By all means call them puppet states, but I want the old game mechanic back. Perhaps a military presence would be required to keep them from revolting. And while you're at it, give me some non-military way of 'acquiring' city-states (like buying them)
I'm indifferent to vassalage, but agree it would be interesting diplomatically. I don't see any need to remove the puppet city mechanic, however, as the two don't represent the same thing and work very differently. There's no problem I can see with the puppet mechanic - again there's just the ever-present AI issue that Civ V AI isn't very good at anything much, including city management.
5. Colonies. I'd like to be able to choose whether a new city I settle is a part of my homeland or a colony. Colonies could function much like a puppet state, except rather than a military presence to keep them in line, happiness would be the governing factor.
I don't see an obvious need for this, or see what it would add to gameplay. It's not a function I recall using much if at all in Civ IV.
6. Religion. I'm not sure anyone could disagree with it returning.
No (although it's a little disingenious to claim that it's gone, any more than you'd necessarily say Civs 1-III didn't feature religion. This is, after all, a game with monasteries, temples, a technology called Theology and a social policy branch devoted to Piety. It's just not handled as a separate mechanic). However, the existence of several threads with multiple contributions giving different ideas of how religion's role should be expanded make it clear that there is considerable disagreement over how it should work, so the designers of any expansion would have a much tougher task than just "bring back religion".
One key element I'd suggest is that a desirable religion mechanic would not be the Civ IV one; it would incorporate similar types of benefit and drawback (principally in diplomacy), but would remove the 'exclusive religions' problem of Civ IV, remove the more game-dominating aspects of religious bonuses (such as linking the most efficient and earliest happiness buildings to religion), and (in keeping with other Civ V changes) wouldn't so explicitly favour wide empires (the more religious buildings/cities of that religion, the more you get out of it), without some corresponding benefit to 'taller' ones.
7. Corporations. Less popular than religion, but to me added yet another layer of depth to the game. Perhaps it shouldn't function in the exact same way, I don't know.
Corporations, like religion, felt very experimental in the way Civ IV handled them, however the idea of a corporation-like mechanic tied to resources was strong, and would probably play well with the way strategic resources now function in Civ V. It would also allow early-game resources like iron to remain useful in the later stages of the game, if some corporations had benefits that played off those (Coal in particular needs some love, since it's currently useful only for factories, and is usually available in sufficiently high quantities that limitation is never a problem).
8. Espionage. Again, provided another layer of depth to the game, and one of the most enjoyable games I had was trying to create an espionage/production cross economy for the first time with Stalin on the Parker World map
Yes, espionage should definitely make a comeback in some form.
9. Seriously penalise huge empires somehow. So many times have I seen a massive AI empire steamroll the map, and they only build up momentum. Probably due to happiness bonuses etc. Huge empires in real world history have always crumbled eventually. In Civ 5, they build up such a tech lead it's impossible to move them.
This is a constant of Civ games. If this is to be tackled the above additions need to be designed with this in mind. Religion, for example, only exacerbates this issue if introduced in the Civ IV style of "all benefits, no drawbacks" production-wise and its emphasis on larger empires (unless the mechanic was adapted so that wider spread of religion had a corresponding negative effect on technological development, for example). Vassal states likewise effectively mean big empires are favoured - both because of the benefits vassals provide indirectly, and because larger empires are less likely to be forced into vassalage themselves.
Edit:
10. Change trading posts back into the cottage/town dynamic.
Would be gamebreaking with the larger numbers of tiles that can be worked in Civ V and the sheer importance of gold.
Changes I'd like to see (other than superior AI) would include greater diplomatic options and situational awareness in city-states (such that their missions are relevant to in-game events), some form of religion and espionage - and possibly corporations - international trade routes restored (at least with city-states), map trading, and probably some expansion to the tech tree, as well as tweaks to Great Scientists and research agreements to limit abuse. I'd also like some change to the diplomatic victory condition, so that each player/CS only gets 1 vote (as now), but there is a classic Civ-style election between the two strongest civs, with other civs and CSes having to cast votes in favour of either rather than voting for themselves by default.