I’ve taken care of my mother for two years now. The doctor wrote a letter stating I’m a full-time caregiver. Is the house safe?

Was there an explanation as to why the NH's application superceded the one being done for you? Bad idea to give that control to them, you'll never see any claims statements.

Medicaid doesn't take the home of a spouse living in the community, can't see that it could take an adult child's. Sounds like your experience with an elder attorney hasn't been a good one. Perhaps check with friends/family for a referral elsewhere.
And if the attorney doesn't give you an explanation and refund, perhaps the State's Ombudsman office can help with that. According to their website, a NH patient (and by extension, their representative) has the right to manage their own financial affairs; that should include filing for Medicaid. Their information line is 800-622-4484, their website is ombudsman.IN.gov.

I think you need the help of that attorney again for these questions.
They are very complex. I don't think a Forum can know all the legal details involved, but I certainly do wish you the very best of luck.
I suspect that Medicaid will not take this home, but according to who is on the deed and title it may have a lien placed on it so that the government gets some compensation for money spent on Mom at the time of sale of the home. The life estate should protect you in your lifetime, would be my GUESS, but again, I am not an attorney.
I hope Igloo is about and may give some opinions, but remember, we are a forum of ordinary folk and these are legal questions.

Medicaid does not take houses. If the NH is telling you this, find another lawyer, an Elder one versed in Medicaid. The house should be exempt till Mom dies. Your Doctor needs to say you have been Moms caregiver for 2 yrs and living with her. Then u will be covered by the Carefgiver allowance. If your on the deed, you should not lose the house. I think your circumstances are too complicated for a NH to deal with. My cousin was told not to let our local NHs fill out his Moms application. If they dropped u and did not do the work you paid them for, ask for your money back.