How much weight can a 3/4" x 8" lag bolt hold when drilled in half way?
How much weight can a 3/4" x 8" lag bolt hold when drilled in half way?


It's not just about weight but where and how that weight is applied. In your case, it would seem that the how is purely vertical, that is, no tension and applied to the perpendicular grain of the tree. The where is still open. Where on the bolt is the load applied? The further it is away the less it can carry, and the design problem changes from purely shear to a moment (tension / compression) situation. 


  What about other lengths and thickness, and percent drilled in?


There is far too much math and variables to answer this question yet. You need to lower the variables first, aka settle on a design and work out some of the details first


  Is there some sort of formula to calculate?


Yes, many of them but these are outside the scope of this website

The description and everything else aside, this all comes down to loads and resistance based on things like material properties and the specifics of your design. Since you yourself are not sure about the details then it's not possible to answer this question yet

In general, lag screws (they are -not- bolts) can be used in place of the tree house bolts as there is little fundamental difference between them and the tree house bolts other than dimensions and maybe steel properties (but that probably may not control design).

But to specifically answer your question, you'd have to hire an engineer, sorry