A: It would be assumed that anything in the house would have belonged to your father and came to you through the probate until proven otherwise. Here it is clear that you know that the tools belong to your brother. I would further advise that tools are not worth causing additional and further family strife over and therefore setting the tools aside for your brother upon his release would be the best course of action. I would further wonder where the probate attorney who did your probate is and why he is not specifically advising you on this matter as that is what he was precisely paid for by doing the probate.
A: No, you as the Personal Representative should handle this account and check writing. I assume there is some sort of a trust issue, concern or worry on both sides here for some reason. A Personal Representative should write out checks as directed by the law offices or as reviewed by the law office if there is any doubt. I will presume that the law office has discovered this is the best way to handle any errant check writing by Personal Representatives based on experience. If you take a poll of probate attorneys, I would presume that most all do not hold the checkbook. I have never held a checkbook for a Personal Representative. The again, it may be the way they have determined is best to handle probate matters.
A: Very sorry for your loss on the passing of your father, please accept my condolences for you and your family at this difficult time. When you have assets with a beneficiary designation (pay on death or transfer on death) the asset passes to that person/s outside of probate, it is not a probate asset. Usually all you have to do is produce an ID, death certificate and complete the bank paperwork.