I think there has been many posts about Bagan that you could check on the forum, go to Destination and Bagan (Nyaung U) subforum and you'll find quite some tips about Bagan: Bagan travel forum
Another one interesting is a bit harder to find since it's in Introduce yourself subforum, but Billy gave some useful indications about the main places in Myanmar: Tips about travelling to Myanmar
Then, as for your specific questions, I think the best place to stay if you want to be close to the temples, is Old Bagan, but then don't expect to have any kind of entertainment, apart from what is provided at your hotel. Then choosing between New Bagan and Nyaung U also depends on what you are searching. Nyaung U you will definitely have the most restaurant and entertainment available, even though it is really not extensive. And in New Bagan, you will be a bit closer from the temples and from Myinkaba (the lacquer-ware village) with some choice for restaurants as well. But don't expect any parties in Bagan at all.
A good point for Nyaung U, is that you can visit the Shwezigon from there, which is a beautiful golden Pagoda.
For souvenir, the must is the lacquer-ware, which can be at times very expensive. Depending how beautiful you want your souvenir to be or who you want to offer to. Sand painting is also an option and you can find it often around the temples.
Horse cart over bike depends on your physical condition and how far you are from the temples. Roads are mostly sandy, so not always the easiest to ride, but for me it is the best option for one day, you can cover a lot of the temples while stopping whenever you want, your thigh will be terribly tired at the end of the day though (cannot cycle two days in a row). Horse cart is a nice way as well, traditional and less exhausting, but difficult to ask the driver to stop everywhere for a photo. Prices are around 3000 kyats for bike (or free in some hotels) and horse cart should be 25000 kyats for a day (8-10 hours max).
Ticket is mandatory and cost now 15 USD, but there is nobody to check it if you arrive by road to Bagan, there are even not any counters to sell these tickets as far as I know. They will do check of the tickets at sunset in Shwesandaw though, and maybe in other popular sites. So maybe you will have to pay, maybe not. If you don't go to popular temples at peak hours (sunset and sunrise to a lower extent) you most likely will not pay. But in this case, just donate to some of the temples, these temples need to be maintained if future generations would like to witness their beauty as well.
How much time do you have in Bagan for other activities? Shopping lacquer-ware can take some time if you want to compare prices and styles.
Some activities that will take longer would consist on going to Mount Popa or other attractions around, but half a day or a full day is needed.
Hope this helps, ask more if you need.