1. 足立: あ、仕事のことは、あの、心配なさらないでく ださい。僕達でなんとかやってますから、これを機会に ゆっくりお休みになってください。
Hello again
This is a continuation of 足立 and 部長's dialogue from yesterday, right? Remember how I said that 部長 was recovering from some health issue?
In the phrase これを機会に, これ is referring to the circumstance that 部長 had fallen ill. So 足立 is saying that 部長 should take the chance while he's not feeling so well to really rest up, instead of worrying about work.
2. 部長: ま、そう言ってもらえると、少し気が楽になる な。じゃ、この際、ちょっと骨休めでもさせてもらうと するか。
Yes, you're correct. ・・・とするか is a way to declare one's intent to do something. 部長 could also say: させてもらおうか and the meaning would be the same.
Note however, 骨休め refers to something more substantial than a "little break". It's actually a pretty good break, something intended to revitalize one's energy. It makes me think of going to a hot springs or something.
ちょっと and でも are used here not for their literal meaning. They're used to make the sentence more indirect. It's appropriate to speak indirectly when you take time off of work for personal reasons.
3. 彼に過ちをあからさまに白状した。
彼に here is "to him", so... (I/he/she) openly confessed their mistake TO him.
4. ああ、あ、いいですよ。僕でよかったら。
What does 僕でよかったら mean? Why で? Why よかったら?
"if (you're) comfortable with (=で) (telling/asking) me" is what's meant here. This is a common expression used to make it easier for someone to open up to/rely on you. You can also say 僕でよければ.
Edit: 僕でよければ also conveys a sense of humility. If you use it before making a request that involves you directly, it conveys something like "I know I'm not the best person for this, but..."
5. それが、時々、彼女何考えてんのか、よくわからないこ とがあるんですよ。
Yes, this is an abbreviated form. The full form would be 何を考えているのか