I think I'll be voting for Elizabeth. I like GP's for early wonders / religions.
I'm not sure that I'd vote for Elizabeth for this particular reason. If you're wanting GP's for early wonders or religions, then you'll need to either (a) get a civ which starts with Mysticism to grab Buddhism or Hinduism, (b) build Stonehenge or the Oracle (for Great Prophets), or (c) build the Pyramids or Hanging Gardens (for Great Engineers). There is simply no other way to get a Great Prophet or Great Engineer in the early game. (I don't think Metal Casting and Forges for Engineer specialists counts as the early game.)
Obviously with respect to (a), Elizabeth does not start with Mysticism, so getting an early religious tech first is probably out of the question. With respect to (b), Stonehenge and the Oracle are certainly cheap and easy to build, but this makes them very popular with everyone. With up to 5 teams interested in building one or both of these wonders, we can't rely on getting either of them. With respect to (c), both the Pyramids and Hanging Gardens pretty much rely on us having access to stone (not to mention being able to afford the time to invest in building them).
Obviously it's up to you which civ you prefer.
I just thought I'd point out that Elizabeth isn't a great choice if you're intending on a strategy of getting early wonders and religions through great people. (An Industrious leader - someone like Huayna Capac - would probably be a better pick for this strategy, because they get an automatic bonus on building any wonders. Huayna also starts with Mysticism.)
Most cities will be building banks.
Perhaps, but not until quite late in the game. (Bear in mind that these democracy games usually only last until the industrial ages.) Thus the significance of the bonus on the Banks will be minimum.
Redcoats could be a late game war clincher?
Redcoats are certainly very nice, although do they come along quite late (with respect to the timeline of a democracy game). Thus I doubt that they'd be much of a "clincher" for wars - by the late game, the civ in first place will almost certainly have a significant lead over the others, and thus the effect of Redcoats on determining the outcome of the game will probably be very minimal.
As a general rule, earlier UU's have more of an influence on the outcome of the game. I certainly wouldn't turn down Redcoats if offered them, but I think that an earlier-game UU might be slightly more beneficial.
I guess on a watery map I'd vote for someone else, Willem maybe.
Willem would certainly be a good choice for a watery map, although I think that Ragnar would probably be even better. This is because Ragnar's UU and UB are geared towards an earlier part of the game than Willem's (and thus they are more useful for more of the game). Also, the Aggressive trait probably provides more of a benefit than the Creative trait in this type of game.
On a rocky map, I'd probably vote for someone else.
Why do I keep thinking about the Mongols?
Personally, I think that civs which have Horse Archer UU's are less than optimal in MP games. This is because there are so many important techs to get in the early game, and Horseback Riding is relatively expensive and a dead-end tech. UU's which become available from techs that you will always want to research anyway (The Wheel, Bronze Working, and Iron Working for example) are more useful, in my opinion.
Bear in mind that my aim is to inform, not to force anyone's hand. Ultimately, what leader you prefer is completely up to you.
Yes, I believe so.
Of course, I'd rearrange a play style to suit an Ottoman civ, but I still think that Janissaries, while powerful, have a very limited window of use - unlike, for instance, Macemen and Riflemen.
