In many states Allstate, and others, have started accident forgiveness plans where your rates don't increase because of a first, or any, accident. That depends on the options you took.
They also have options where your deductible is lowered each year you go without an accident.
If your insurance company pays your claim, they should go after the other party for what they paid, and your deductible, if any.
In many states, as long as they can recover a set percentage, they cannot raise your rates. In NY they only need to recover 30%.
But all may not be as rosy as it sounds. Most policies provide good driver discounts. I'm not sure if you lose that for any accident. I always felt that was their way of skirting the rule that says no surcharge. "We're not charging you more, just giving you a smaller discount.
Just like the owner's manual, everybody really should read their insurance polices. They aren't that long, and anyone with an 8th grade reading capability should be able to understand it. The days of all legalize are gone.