3 or so months in the USA

[batmanreturns]

Our two year old daughter has a very rare lung condition and during the UK winters she struggles massively with bugs that a normal child would fight off with a bit of rest. Therefore we are looking to relocate during the winter months to the USA, (suggestions welcome on somewhere that has a high temperature at that time of year in the USA?) so that she avoids the usual UK winter bugs and she gets a chance to develop without set backs. She is oxygen dependent so we will need to order liquid oxygen when we are out there (again, any contacts would be great as at the moment we are going round the houses trying to find someone to supply it!)

My questions are as follows if someone can help me out as we are struggling on where to start at the moment.;
1. Can we stay in the USA on a tourist visa for 3 to 4 months?
2. Does anyone know of any good property sites to find houses to rent?
3. Transport-would it be best to rent a car long term or buy one wisely and resell when we leave?
4. What is the real world temperature during October to February for example?
5. Any other issues we need to be aware of or plan for?
6. My business is remote so i guess there is no issues working from the USA on a firm that is based in the UK providing i'm happy working on UK time?

Thanks in advance for any assistance.

[batmanreturns]

thanks all.
So, if i don't work from there and we only stay for 3 months then if i'm reading it right, it should be ok?