Rental companies typically do not check your driving record. While some may scan your license when you check in, it is usually to determine whether your license is valid.
You are typically covered to a certain degree under your current car insurance, but you may want to compare coverage from the rental car agency, as well as your credit card company (the card you used to book the rental car). In some cases, you can upgrade your coverage, which will protect you in extenuating circumstances.