Try re posting your images

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Not sure why the direct links from my ISP are not working.
They are being clipped off.

I am adding a link to photobucket

Here is a link that might be useful: Small Table

This looks like an interesting project. The burn mark looks too deep to remove without replacing the veneer. The bubble is straightforward. With a razor, slice the veneer with the grain the full length of the bubble and with a syringe apply glue under the entire bubble.

But before doing these repairs the table should be fully stripped and the wood cleaned. What did you use to strip it? Sometimes this takes many applications. If the piece can be disassembled easily, it should be done. Check all glue joints to see if any are loose or can be loosened. If the glue joints are all tight, leave them, but stripper sometimes loosens them. Depending on where you are in the process and what chemicals you've already used, will determine what steps to do next.

You'll need to have a number of solvents on hand, but you should post the details of what you've already done before going on.

I did glued and fixed the bubble.
I did a light sanding and the burn mark will not come out.
I stripped with Zinsser Power Stripper.

Not sure what the green color is on top. It does seem to sand off.

The piece is solid and does not wobble. I am not going to take it apart. It appears to be glued together. I cannot see any screw or nail holes.

Not sure if it is actually old or not?

IMO, only the center panel is mahog (veneer, at that), the rest all stained mixed hardwoods. It was never intended to be without its blood-red thick polish finish. I would think it was made well within the last century, 20's to 50's.
The black stain looks to be a chemical reaction, nothing that can be removed; it's like a permanent dye in the wood, goes all the way thru the veneer.
Casey

All done, doily will cover up the burn mark.

Looks good for free furniture!

That's PRETTY!