Does the ability to write both the traditional and simplified version of a character count as two characters in character counts?

Also, on our profile it would be cool if it showed if we were learning traditional, simplified or both.

Two. But if the character is the same in both 簡體字 and 繁體字, then it will only count as one character. So if you are studying 一個 in both simplified and traditional, your character count would be three: one for 一, one for 個, and one for 个. At least, that is the way I understand it.

Interesting, that might explain some people's character counts being so high. Perhaps I am not as far behind as I thought then...

If a character takes different tones in different words, does that increase the character tone count? Apparently I know (and have added) more character tones than writings.

No; it's just when there are multiple traditional variants for a character, and those variants have different tones.

There may be some bugs with the tone counting, though, since that doesn't explain the variance that people generally see.