1) What does "spaced" mean in "Dude, you know you have been really spaced lately"?
I can only assume as Rhiannon did that it is short for 'spaced out'.
2) What does "drop the sack" mean in "I dropped the sack and straightened up"?
It is probably literal. He was carrying a sack and put it down.
3) What does "bottom out" mean in "my heart bottomed out"?
'Bottomed out' means arrived at the lowest point. I assume it means the speaker became as sad/depressed/lovesick as they have ever been.
4) What does "tackle" mean in "he tackled me with a hug"?
He came up at speed and hugged me vigorously. The hug was almost like a tackle in rugby or american football.
5) What does "spitty burst" mean in "he cried in spitty bursts"?
I see it as crying out loud - something like 'blubbering'. Spit was occasionally coming out his mouth as he was crying.
In all of these, more context helps. With such a small amount of context, it can be hard to know what is meant.