1) You would need to prove there is a danger. Typically speaking trees need to have a trunk lean over 10 degrees and or some signs of root instability before there is a danger of uprooting. A large protractor can help you measure this. You could also bring in an arborist for evaluation. Tree companies typically do terrible evaluations as tree removal is the service they are selling. Also they like to top off trees as it creates a never ending pruning loop.
2) Nearly all claims increase your premium. It will probably go up for a bit and back down for about the cost of a tree removal.
3) Don't know, that would be clearly stated in your insurance police. Speaking of why don't you just call your agent or the insurance company. They would probably be way better at this than an inter-web stranger.
4) No HOA board is going to deny an arborist telling them to cut down the tree. They wouldn't want the liability nor would it hold up in court.
Kindly post photos of the tree, take measurements, any additional infor (species, soil type, etc.) if you would like further evaluation.
For reference:
Tree that needs removal: https://cdn-s3.si.com/s3fs-public/golf/2014/11/29/gallery_images/oak_hill.jpg
(Note the sharp lean and bulge from the beginning of uprooting)
Normal trees: http://c8.alamy.com/comp/A0H4EN/black-oak-tree-in-autumn-in-high-park-in-toronto-ontario-canada-A0H4EN.jpg
(note the left most tree has a good lean, but would likely be evaluated as normal. Oaks tend to grow into the sun in this way and are of very little risk.)
TL DR: Get an arborist to make an evaluation and call your insurance company.
