It is often the case that the successor trustee is one of the beneficiaries. The successor could even be the SOLE beneficiary. The main time you would NOT want a trustee to be a beneficiary is when the trust contains provisions staggering the distribution of assets to the beneficiary. In that case, it would be like the proverbial fox guarding the hen house.
As far as the mortgage is concerned, you do not need to direct that it be paid. Whoever succeeds your interest in the home would inherit the property subject to whatever liens exist at the time of your death. If that debt is not paid, the lender would be within their rights to foreclose. So the debt WILL be paid whether you specify it or not.
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Yes; and you needn't require payment of the mortgage. That is a natural consequence or the property would be lost to foreclosure. Although it may be more complicated, so would merit some mention in the Trust.
If I am understanding the question, I think you are asking about instructions for the time period AFTER the beneficiary receives the property from the trust. If that is the case, instructions are most likely not needed because he would know that he needs to take over making the payments in order to keep the building from going into foreclosure.
If you are asking about the time period BEFORE the beneficiary receives the property from the trust, but while he is managing trust assets as the successor trustee, instructions are most likely not needed because again, the successor trustee has a duty to maintain and preserve trust assets, which means paying the bills.
In fact, if you mandate that the successor trustee must continue to make payments on the property while it is in the trust, in today's real estate market that may cause a problem because we are seeing cases where it is more beneficial on an underwater property to stop making payments and let the property go than to keep making payments.
You should make sure that your successor trustee knows where your important documents are kept and that they will have access to mortgage bills, etc.so the transition of all financial information and responsibility will go smoothly.
That being said, your question also sounds like you are drafting your trust on your own without an attorney's assistance, otherwise your attorney would have been able to answer this type of drafting question. While this is a worthy goal and many people have access to documents, there are many cases where a mistake in the documents has cost the survivors many times more to fix, than it would have cost to have the estate plan properly prepared by an attorney in the first place. At the very least, please have your trust reviewed by a trust and estates attorney to make sure it will accomplish your goals. An estate plan controls what will happen to your entire life's savings and a trust could be the single most important document you may sign in your lifetime. It is also part of the legacy you leave behind. This is not something you want to take any chances on.