#1
I have conducted a study where participants are split between two groups (treatment and control). I want to measure changes in depression and sleep disturbance pre and post-test (before and after intervention). I want to see interactions between-subjects for depression and sleep. I also want to see interactions within-subjects before and after treatment for both depression and sleep.What analysis would be appropriate? Would I do multiple analysis?

Depression Sleep

Treatment Before After Before After

Control Before After Before After
 
#2
For each dependent variable you could perform "mixed" analyses of variance, including
a between-subjects factor (group) and a within-subject factor (time). This would tell you
something about whether depression (or sleep, respectively) changes over time, and
whether change is different between groups.

But I am not sure what exactely you mean with
"I want to see interactions between-subjects for depression and sleep. I also want to see interactions within-subjects before and after treatment for both depression and sleep." If you want to see interactions, you need 2 factors
which interact. Do you mean something like "I want to see whether
change of sleep over time is different from change of depression" ?

With kind regards

Karabiner