Hi Kayad,
Hard to advise, but I can share our experience with filing a homeowner's claim for water damage.
We returned from a weekend out of town to find that our dishwasher (which we had started as we were leaving out the door) had somehow leaked onto laminate flooring we had under it. Only a couple of pieces looked wet/warped, and I assumed it was no big deal. My husband did some googling, disagreed, and started pulling up a few of the laminate pieces to find some standing water underneath. We called insurance, I figured they'd just put some fans up and try to match the laminate, but they sent a remediation company out and those guys immediately removed every piece of laminate until they stopped finding any "signs of water". That ended up being about a 10 x 10 foot square of our kitchen and dining space. They put out fans and dehumidifiers, then came back and rolled out cardboard for us to temporarily walk on (not ideal in a kitchen) while we dealt with insurance. Insurance paid the remediators.
Long story short, insurance ultimately paid for us to replace ALL THE FLOORING in downstairs, because the laminate had been "continuous" throughout the entire first floor of the house. So they allowed us to pick a new "similar cost" flooring to put in. (We actually took the opportunity to upgrade to porcelain wood look tile, because its indestructible, but we paid the small difference in material and they still paid the extra labor for tile installers rather than laminate installers). They paid for new trim along the floor of the entire downstairs (since the original trim was removed for tile install). They also paid movers to have our things packed up and moved to a storage pod in the front yard (which they also covered) while the flooring was being replaced, and paid for movers to unpack the pod once the work was done. They paid for house cleaners to come and clean the place top to bottom after all was done, and paid to compensate us for extra food costs while we couldn't access our kitchen.
What was initially something I thought I couldn't be bothered with (and looked so minimal) ended up being a multi-week ordeal, but we got brand new floors with minimal negotiation.
To your situation:
1. Yes to emphasizing the "sudden and accidental" nature of this incident. That is key insurance speak for "are we going to cover this or not". If water damage is caused by an ongoing leak or drip that the homeowner knew about but ignored, its not covered.
2. After my experience, I'm pretty pro filing a claim. I had no idea how damaging a small amount of water could be and never would I have thought the water had extended as far as it had - there was almost no visible signs that it had.
3. I'm sure our insurance premiums went up a little, but not more than $100/yr? We have stayed with the same company and this incident happened probably 4 years ago now. No price gouging that I can tell.
4. I advise you to ask your insurance company if you can call your own contractors vs. using the ones they automatically dispatch. The people they contract with are incentivized to do the cheapest/simplest fix. I wanted an objective opinion and called 2 or 3 local contractors to get their quotes in addition to the insurance-suggested person. Our insurance company had no problem with this, and accepted the quote we chose and submitted.
5. I did not sign anything or agree to the final amount until everything was done and dusted. The insurance agent wants to close and resolve the claim as soon as possible, but I wanted to make sure the work got done before I closed out the claim. Again, they didn't give me much push back. But just be aware that if you do file a claim, don't sign an agreement about a specific dollar figure until you know what the repairs will cost fully. Be prepared to advocate a little if need be. I had to ask for the insurance to cover movers to get everything out in order to install new floors - and they were agreeable to it, but I had to ask. It wasn't automatically offered. So think through all the associated costs with any repairs and be sure to ask for those to be covered.
Obviously, my comments relate to my experience. I'm not sure which company you're with, but assuming its one of the big players, I assume they would be similar.