What is the distance between the end of the bed and the edge of the drawer? What is the weight capacity of the bed? Does it require/allow for a box spring? Are there drawers on both sides of the king-sized bed? Are the drawers on casters or on pull-out rails?

What is the distance between the end of the bed and the edge of the drawer? around 2.5"
What is the weight capacity of the bed? which size are you inquiring about?
Does it require/allow for a box spring? does not require a box spring 
Are there drawers on both sides of the king-sized bed? 1 side only
Are the drawers on casters or on pull-out rails? on casters
Here are the assembly instuction links
https://www.titusfurniture.ca/wp-content/uploads/2021/03/R189.pdf

https://www.titusfurniture.ca/wp-content/uploads/2021/03/TRUNDLE_1.pdf

How thick a mattress will the trundle drawer accommodate?

The mattress thikness for the trundle should be less than 9" to accomodate it best.

Hello, does this beds trundle pull out on either side?

Yes, you can put it from either side

On a full size bed, is the size of the trundle a full or twin size?

The Trundles are always Twin size