The handling of claims and convictions is very specific to each scheme, do you target people with them, do accept that a few are okay or do you look for perfection.
Premiums will be affected by both the conviction code, such as SP10 (speeding) but by the number of points, the fine and if a disqualification resulted. All of these give an automated system an idea of how bad the offence was.
Whilst some offences are accepted almost as a fact of life, a simple speeding conviction will cause very few schemes to not want you. A drink, drug or non insurance offence is the opposite and you will be left searching for niche schemes targeted at this type of offence.
Claims too have some types that are almost ignored such as windscreen claims, they are often deemed to be a fact of life and not your fault. Accidents and the costs associated with them are loaded in relation to their costs, with Malicious Damage sitting the middle in the it depends category.
It is important not to "forget" to mention claims and convictions, not only are there numerous databases that can help and insurer detect these omissions, doing so could see you on a fraud database making getting insurance very difficult.
When you go into the broker or use an online site, these databases probably won't be checked, the quote process needs to be as fast and as cheap as possible. What happens after you take out the policy would be insurer dependent.