In regard to the bill, while you were not executor of your father's estate, you were an heir
. In addition, you are now in charge of your step-mother's estate and need the same information she needed to proceed to pay bills on your father's estate and on behalf of his estate. It would be highly unusual to have to reopen his probate and get someone appointed as executor just so you could obtain copies of billing statements regarding administration of his estate. It sounds like they're just trying to make things hard for you. Getting your new attorney involved, once you have one, should make that move forward so that you can obtain copies of the billing statements.
As far as getting a copy of the file. There is no time limit, but a reasonable time to get a file together should only be a couple of weeks at the most unless you're talking about a very complex probate that's been going on for years -- that's not the case with your step-mother's estate. Again, getting your new attorney involved may help to speed up their compliance with your request for the file. The problem is that if you still don't have it within another couple of weeks, there's very little that can be done that won't take a substantial amount of time to produce results. One of the best things you could probably do is threaten a bar complaint for failure to turn over the client file upon termination of representation. That is serious to attorneys and something that is a violation of the ethical rules, so that might be something to consider.
Finally, as to billing the estate for meeting with someone that was threatening to sue the estate -- that's normal because attorneys often do meet with these types of peoples (on behalf of the estate) and explain they will not settle and that they can sue if they want but they won't get anything. It is a manner of effectively defusing matters before they become a large problem.
However, it is highly unusual that you were not informed of this person's threat of suit and asked if you would like the attorney to meet with this person to handle the situation. For them to just meet with the person without your knowledge and without getting your approval is not proper and I would contest that charge if I were you as an unauthorized action by the attorney.
Please let me know if I can provide clarification or additional information. Thanks.