Answer to #1 - read either Rash"i or Rada"k who agree on a general theme regarding each of the people David mentions. He states that Shlomo should not kill them immediately, but each of these people, by their own behavior, will eventually perform some action that requires being killed. In fact, each of them did just that, eventually.
The general point is that each of these 3 people did something at the original time that was rebellious against the king. The king is allowed to kill someone who rebels against the throne. For various reasons, David did not do this, so he commands Shlomo to "finish business."
I am surmising, though, that a new king, perhaps cannot execute someone who was rebellious against a previous king and was not executed then. I.e. - the execution status is not "inherited". (Something I have to check in Ramba"m). Regardless, see the above explanation, which was David's intention.
Answer to #2, heard from my Rav when we were studying Melachim:
David didn't mention that Yoav killed Avshalom because that was actually a benefit to Shlomo, at the end. Had Avshalom not been killed, either he may have succeeded in killing his father (David) and / or usurping the throne, then (which was his intention), or he would have tried doingthe same with Shlomo, now. Either way, it would have been less likely that Shlomo would have been king.
The point is, that David intentionally, didn't mention Avshalom being killed so as not to put any positive thoughts in Shlomo's head, at least not directly. (We can assume that the wise Shlomo, knew this angle, anyway.)
I believe that Rash"i may mention this aspect as well.
For #3 - I have to explore further. Offhand (Guess), there may have been a tactical reason to spare Yo'av at the time. I have to read Shmu'el, again, to recall what happened.