Does his daughter have any rights to personal property purchased (In his name only) during our marriage? i.e. cars, firearms, bank accounts.
Family Law Attorney
Generally the personal property passes to the surviving spouse. You need to check how the bank account were titled and if they had a payable on death designation for someone other than the surviving spouse.
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Real Estate Attorney
If he died without a will, then the statutes set the rules under what is called "intestate succession". This covers assets that do not otherwise pass by other rules. For example, bank accounts, IRAs, and life insurance may have beneficiaries named or joint owners named. Those items pass outside of probate. In other words, no one here can say whether his daughter has any rights and you need to sit down with a probate attorney and go through the documentation of your husband's assets to get a complete answer. It is most likely that the assets would pass to you as the surviving spouse, but you need to examine the specific assets to be sure.
Disclaimer: This answer is provided for informational purposes only, does not constitute legal advice, and does not create an attorney-client relationship. I am licensed in Arizona and can only provide general comments on matters outside of Arizona law. Actual legal advice can only be provided after a direct consultation in which all of the relevant facts are considered before providing a response.