Should I be listed on the deed of my father's new house or put the house as an inheritance in a trust?

See an elder law attorney.

I agree with AlvaDeer about seeing a lawyer.

They can help your father avoid unintended consequences with property or capital gains taxes, gift tax return (but almost certainly no tax), personal liability, creditors, and Medicaid look back.

You can start doing homework now by looking at web sites like https://www.elderlawanswers.com/is-it-better-to-use-joint-ownership-or-a-trust-to-pass-down-a-home-15966

I also recommend a book on avoiding probate from NOLO Press, often available from local libraries in print or electronic format.

There is no capital gains tax on profits of less than $250,000. There are many ways to transfer ownership upon the death of the owner (dad). Both transfer on death (TOD) and JTWROS can state multiple survivors. Both would avoid probate. Both also supercede a will. With TOD you and brother would automatically become owners on the death of your dad. With JTWROS all three of you are 100% owners immediately. Either would have to be listed on the deed. Both have pros and cons. Consult an attorney to decide which option would be the best.

JoJo, best for your father to speak with an Elder Law Attorney. Said attorney may suggest the house be placed in a Revocable Trust. Note that State laws do change over time and the Attorney can keep your Dad up-to-date on those changes so the Trust can be updated.

It is never a good idea to be place on a Deed to a parent's house. Otherwise, you would be responsible for Capital Gain Taxes when you sell the house, and the bases used for the Taxes will go all the way back to when your Dad had purchased the house. It is better to inherit the house, that way the Taxe bases will only go back to the date of inheritance.

Also, make sure the full address of Dad's new house is placed into the Power of Attorney for later down the road, in case Dad is unable to do the paperwork for selling. It just makes everything much easier.

You might get hints and clues here, but in the case of a home and its protection always go to a lawyer. The only way to avoid probate, if there are any assets beyond almost nothing, is to create a Trust. Trusts do not go through probate. However, if a Trust is created it is crucial to be certain that properties and accounts that one wishes to go into the Trust does indeed get titled in the name of the Trust via the courts. This is nothing to seek amateur advise on. Many states now are closing the loopholes that protect inheritance by Trust. California just did so in a vote last election. As you can imagine, passing a four million dollar home, held by an elderly couple for many years with prop 13 low taxes, passing on to heirs in a Trust was losing a whole ton of money to the State.
So again, you must seek the advice of a Trust and Estate Lawyer. These things have huge learning curves. When my bro wanted me to manage all his finances he thought it was best to put me as his POA. But we soon enough found out that with his Trust, and his real property, some of his accounts, and his SS automatic deposit going into the Trust it was crucial I be also made Trustee of his Trust. So these things are complicated. You are dealing with some money. I encourage you to get books on Trust and Estate, learn as much as you can, get a lawyer for this stuff. As they always say "it's complicated".

My dad deeded his home with a life estate deed. The home was completely his during his life and transferred to me upon his death. This is commonly done to avoid probate, every real estate and lawyer I’ve dealt with was familiar with it