how much of the seat's tube should be inside the frame?

Imagine the frame being transparent.

The bottom edge of the seat post should be about 1 cm (1/2") below the bottom of the top tube. If the seat stays are below the top tube then the seat post should go below the bottom edge of the seat stay.

If there is a minimum insertion mark on the seat post, use that instead.

Most seatposts are marked in someway to show you the minimum insertion allowance. I believe it is approximately 4 inches.

A large frame is probably perfect for you. I know several MTB riders whose measurements say they should be on a large frame bike, but they prefer to ride a medium. Don't be surprised to find that your bike handling skills improve.

Bikeworks answer is excellent, he gives you 2 different ways to figure out how much of your seatpost should be inside the frame. Measure both ways, and if you have any doubt, you might invest in a longer seatpost. You are not overly heavy, but if you are an aggressive rider, err on the side of caution. Best case scenario if your seatpost fails, you need a new one. Worst case, either you damage your frame or injure yourself

The seat post should have a warning marker on it. You may have to look closely and/or wipe it down with a clean rag to find it. But every bike I've ever seen has one.

There should be a minimum insertion mark on the seat post.

they are usually marked

4-5 inches