The LTE choice seems fairly clear as the Spring and Verizon models match the LTE coverage that EE has in the UK. O2 isn't listed, but you can see the information yourself here:
Further information is available from the iPad specifications page - but the GSM/EDGE and UMTS/HSPA+/DC-HSDPA capabilities of both models are the same. The Verizon/Sprint models have more LTE bands and CDMA as well but in Europe, the LTE situation isn't totally clear yet.
If you get an Apple ID in the US - you will be restricted to the content in the US store. The same for choosing the UK store. Since it's a hassle to juggle accounts and you can't get updates from an old country if you move accounts to a new one - it's better to pick one account unless you intend to maintain and juggle both going forward perpetually.
I would buy from Apple if you can since their holiday return policy is very lenient and you could return things in January if you choose poorly or learned more after buying. Also consider, AppleCare is tied to the country where you buy so if AppleCare plus isn't available in the UK and you are OK shipping it here for service, you might let that affect your buying choice if you want that coverage over the standard AppleCare.
The good news is you can easily use her UK account here in the US and all her apps will work on the US device. The only problems would be radio (not a problem on the mini hardware) and warranty support. Since you live here, you can get her a pay as you go month of data with no commitment or long term contract. She won't need to unlock it and can work with her choice of carriers to serve it in England. No carrier lock, no hassle, no jail breaking needed.