Sure, that is done all the time. You may face additional hurdles if/when you do a refinance down the road, as home will likely have to be taken back out of trust and title/lender will need copy of the trust.
Yes you can do so with ease. A trust is usually revocable prior to any of the trustor’s passing and as such is a non-juridical entity. Once someone passes the trust becomes non-revocable and then you have some hurdles transferring property in and out.
When a trust in still revocable and you are refinancing a primary residence you don’t even need to change the title to the property out of the trust. Fannie Mae has guidelines for dealing with trust titled properties, including guidance for signing the documents, can be found on it’s website efanniemae.com.