Do I pay a tax on funds distributed to me from a trust to my sub-trust in Kansas
?

Probably not. Develop an email relationship with the person in charge of your trust, who will know what to expect at the end of the year regarding taxes. Generally you won't pay taxes on distributions.
Study the terms used on your monthly statements and ask what they mean.

Depends on the type of trust and the terms of the trust. You need to discuss this with your trust administrator.

All good advice. YOU probably will not pay tax, but the trust MIGHT!