A: I very much doubt Louisiana would require you to reside there in order to sign the affidavit. Notarizations are always completed in the state where the signer is signing, which most often is their residence.
A: If the vacant lot is titled in the name of the trust, there is a very specific process that must be followed that complies with the terms of the trust with respect to incapacity of the trustee. If the vacant lot is in the name of the individual trustee who is now incapacitated, then the son will sign as agent under the Power of Attorney. That POA document must provide specifically that the son has the authority to engage in real estate transactions. That POA must also be recorded along with the deed transferring the property.
I would not recommend you try to do this yourself without the assistance of an attorney. Whenever you are dealing with real property, the paperwork must be perfect. If it is not, you will have problems transferring good title. In other words, you create a more expensive mess to fix than if you had just hired an attorney to do it correctly in the first place.
A: While a California trust does not need to be notarized, it does need to be signed. However, you seem to have evidence that in fact a trust document was signed (notary records) and the 2016 trust was funded with real estate. Depending upon a lot of other facts and factors, it is possible a court petition might avoid probate. An attorney would have to review everything and evaluate.