Learn English – Where does the phrase “balls to the wall” come from

Tit and tat are used here to mean striking a light blow, so the phrase has exactly the same meaning as blow for blow. They were used as both nouns and verbs, as a sixteenth-century rhyme shows:

Come tit me, come tat me,
come throw a kiss at me.

(Source; An earlier variation has halter instead of kiss.)

I always thought it was a cute mispronunciation of this for that, but apparently it's actually a corruption of tip for tap (Etymonline; concise etymological dictionary), with the same—and, I should think, slightly more obvious—meaning.

This one is actually quite straightforward. It alludes to going out on a branch of a tree. Etymonline says that the figurative sense is from 1897. The Phrase Finder supplies a quote from 1895:

The first uses of it in a figurative sense, with no reference to actual trees or climbing, come from the USA at the end of the 19th century. For example, the Steubenville Daily Herald, October 1895:

[...] If we get the 14 votes of Hamilton we've got 'em out on a limb. All we've got to do then is shake it or saw it off.