I think you'll need to file a petition to force an accounting, return of any missing assets and surcharge for their use of trust assets without compensation, along with suspension of their trustee powers. Trustees have a duty to avoid self-dealing and also to work with each other. Situations with a beneficiary in trust property can take a while to resolve, as people don't like to give up "free" rent, so I'd recommend acting soon.
Probate is complicated enough that I'd recommend using an attorney--even attorneys not familiar with probate court can find it very confusing.
You can check with the county court (in the county where her mother lived) to see if probate's been filed, or if the will (if any) has been lodged. You should also check the house title to se whether probate is necessary, and also to see if the sister has already been involved in title transfers.
Attorneys often don't charge for the first consultation-...
You can indeed probate the copy, and it's common to do so--you'll just need to overcome the presumption of revocation (because the original is missing). If there's no one objecting, it most likely will be accepted at the first hearing. I'd recommend seeing an attorney about proving up the lost will, just to be sure you can meet the requirements. Otherwise, it would go by the previous will (if any) or intestate.
It is most likely a gift. There are complicated rules to follow to be sure she's eligible for Medi-Cal, so see an attorney experienced in Medi-Cal law before making any gifts--it's well worth the money.
Yes, with some conditions given in California Probate Code section 249.5. If your husband specified in a dated and signed writing that his genetic material could be used for such a purpose and he specified who could control the use of the genetic material. Also, the child must be conceived within two years of your husband's passing away.
Check the trust instrument. If it allows compensation, but is silent regarding the amount, you're entitled to "reasonable compensation." You can use professional trustee fee schedules as a guide, but it's generally about 1% of the trust asset value each year.
The trust should have been funded and administered, though it may not matter unless it adversely effects the beneficiaries or causes an extra tax burden. You're entitled to information regarding the trusts under Probate Code section 16061, so that may be the best place to start, and I'd discuss this with an experienced probate lawyer to know your prions and the appropriate deadlines for doing something if you need to.