Does you car insurance company raise your premiums at will?(both)

They increase based on so many factors not just the claim. If they have received more claim on that car in your area they will increase the prem under that post code for all customers. Where you park and drivers is the integral part while calculating premiums, vehicle value age and model is another.
Since you have switched providers read their product disclosure as to what's 'not' covered.

Yes and if you don't claim it isn't that bad