An insurance company's job is to pay as little as possible. It depends on the company, but some of them will comb through a thousand pages of info looking for one tidbit that can let them deny your claim. Others just pay up.
My best ever was Horace Mann. They were expensive, but not terrible. Someone broke into my F250 and stole my leather jacket, the stereo, and some CDs. I assumed the jacket wouldn't be covered. I went ahead and spent $40 at a junkyard for a stereo and the little trim strip above it that was damaged. They sent me a check for $1400: $1000 for the damage, and $400 for CDs and the jacket.
Worst ever was Progressive. My wife's Merc took heavy damage from a hailstorm. Every panel looked like a golf ball and both windshields were smashed. Shop estimated $8000 damage which would have more than totaled the car. Progressive denied the claim because I reported that the damage happened at 6pm, but the NWS said the hailstorm came through at 6:20. I lawyered up and called all of my friends who sent over their claims to show that they reported theirs at 6pm as well. No dice. Progressive cheated me out of $8000 with a massive law team at my side. Then my Impala was broken into and the stereo stolen, window broken, dash destroyed. Progressive denied the claim because I said I was over at a friend's house when it happened. They had to verify my story, and the friend had moved away and I didn't have contact info for him. They cheated me again. There is a reason Progressive can afford those massive national "Flo" ad campaigns.
I was hit by another driver a few years ago and he was with State Farm. The speed and efficiency with which they paid me was unheard of. I had a shop do an estimate because the accident happened in a dealer lot ($2100). Figured I might as well while I'm there. On the way home, State Farm called me to set up an appointment, and he was actually there when I pulled in my driveway. He looked at the estimate, looked at the damage, and wrote a check on the spot for his estimate of $2500.
Needless to say, I switched to State Farm the next day.
As others have said, the deductible works more like a medical co-pay. Every time you make a claim for repairs, you pay the deductible each time just like you have a $20 co-pay for each doctor visit. Each insurance company is a little different, but you can expect to see your rates rise more from the at-fault accident than you will from filing a claim.
But most insurance companies will raise rates after a claim. State Farm tends to take a more "human" approach - I think mostly because they are independent local agents instead of computer algorithms. You may find that making a claim for something like a tree falling on your car won't raise rates because you weren't the inherent risk, but a moving accident (even if it's not your fault) can raise rates. It has to do with your risk. If you have 20 accidents that aren't your fault, the implication is that you must be driving in a way that isn't defensive or causes risk.