1. What is the #1 priority for 2nd/3rd cities: Resources, location or terrain?
Strategic resources are the primary consideration if my capital doesn't have any.
Otherwise I'll mainly ensure a city can be turned into something useful (production, GP farm or commerce-based city) without jumping through hoops. Very low-food cities for 2nd or 3rds are a big no-no to me, as are cities with very few workable tiles. I may still settle such spots later but certainly not for my first few cities.
Unlike the AIs I don't bend myself backwards either to get as many resources as possible in my BFC. I prefer to split resources across 2 or 3 meaningful cities rather than have a single city with 5-6 resources but of no other use due to, say, lack of food.
Hills for extra defense, fresh water access and trees are only minor considerations to me. If I can reach those goals by moving 1 square and not lose anything important in the process I'll do so but that's about the extent of it. I don't like the blue circles in this respect either, they tend to prioritise fresh water access and hills defense far too much IMO.
2. Distance from capital? I tend to try and keep the initial borders around 3 tiles from capital. Too close, too far or disregard for another reason?
Depends a bit on map size this one but generally I'll settle at most 5 tiles from my border, unless there's a very good reason to do otherwise. Examples would be strategic resources again or sealing off a large section of land.
3. Type of city? I tend to look for a production (military) city but I think I tend to force this one.
I like to look for good production sites as well simply because even a single one of those can be a great boon (other cities can focus on buildings a bit more) and they are often hard to locate. The combination of sufficient hammers and enough food to work them is a rarity on some maps.
Overall, I want a purpose for my cities regardless and want to ensure I have either sufficient hammers for later-game production or lots of potential food surplus for whipping. This last point is something the city location algorithm is particularly bad at to my tastes.
4. Do you ever wait and place your capital in a different location than the initial starting point?
I'm not sure whether you mean moving the initial settler before founding your capital or relocating the palace elsewhere during a game.
I'll do the former on occasion simply because the city location algorithm has a few odd glitches.
Moving the palace is something I do fairly frequently. I find that the distance from capital can make a substantial difference during the course of a game, esp. on larger maps.