Yes, but you need to be clear about "personal use." This means that you are converting the LP's into MP3 files only for your use and enjoyment---and not for the purpose of distributing or sharing the music with others. If your purpose is to resell copies of the music and/or even if your purpose is to distribute copies for free, this might still constitute copyright infringement.
Yes. What you can't do is make copies of copyrighted music for any non-personal use. Please see the RIAA's link below.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.