Crank pullers are fairly universal, although there are exceptions (mainly some older french and italian stuff) but you should be OK with the one you linked to. As for the BB, the diameter isn't 68mm, because 68mm is about two and three-quarter inches, which is considerably bigger than that shell. The shell is probably 68mm wide. The crank appears to be a square taper crank.

The second BB you linked to is a) for a 73mm wide mountain bike BB shell and b) appears to be some sort of Octalink one, not a square taper, so it won't work, but the first one might. There are two main things to get right with square taper BB spindles, though: the length and the taper. Shimano use the Japanese Industrial Standard taper, and the other one is the ISO taper, I don't know what your crank is designed for. The two are fairly compatible with cranks designed for the other standard provided you correct for the fact that the effective spindle length is different, but it's generally a better idea to get stuff which matches.

What's wrong with the original bottom bracket? If it's wobbling, it can be adjusted, as it appears to be a cup-and-cone unit rather than a cartridge one like you're looking at replacing it with. You might want to clean it out and regrease it, though.

As a final point, the method you've seen for removing cranks is generally only used when the threads for the crank puller have stripped, but I've heard of it working reasonably well.