I am very sorry for your loss. I agree with my colleagues. If filed within two years, you may qualify for a simplified form of probate called an "informal" probate. However, you may not need to file a probate depending on the amount of assets and how accounts and property is titled. Consult with a local probate attorney for assistance in determining your options. The Chapter 13 should not affect probate issues, but you may have to amend your Chap 13 plan. Be sure to consult with your BK attorney.
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There is no specific deadline. However if it is not done within two years, you need to have a hearing and do a formal probate. Also, a creditor has priority for appointment if no one else has applied to be PR within a certain time (I believe sixty days).
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There is no deadline. However, for a probate brought two years after death, no claims will be presented against the estate and the personal representative can only possess the estate long enough to confirm title in the rightful heirs. An informal probate is available.