I'm utterly uneducated on what you're going through but do want to wish you well in getting help for depression and further insight into what went wrong on ICU.
I have to assume that Attendings and PDs need to feel confident in the medical decision making and thought process of those so close to practicing unsupervised. It's entirely possible that everyone is under the microscope and you're perceiving it more acutely due to being depressed or being in remediation.
1. You can take a medical leave for your depression. Exactly how this will be addressed depends upon your institution's policies. If you have paid medical leave, then you will be paid for your time off. If not, then you can take FMLA unpaid time. Either way, they are required by law to give you your job back after you are better. It's very important to do this before you are "failing out" -- if you try to take medical leave just as you are being fired, that causes big problems. The amount of time depends upon your institution (but FMLA is 3 months).
Unfortunately, you probably would have been better off taking leave before this all becomes a crisis, but what's done is done.
2. It can be difficult to be in a remediation program because of concerns of increased scrutiny. It's a double edged sword -- on the one hand you want to be treated just like everyone else. On the other, you want frequent honest feedback -- which often can only be obtained by telling people that you're in remediation, which triggers increased scrutiny.
3. The needlestick should have been reported when it happened. Now you have two problems -- 1) you have been exposed to Hep C and need to be tested; and 2) it's probably a violation of hospital policy to not have reported it when it happened (which may worsen your situation). I would recommend you report it immediately. There is some chance that your medical insurance will refuse to cover any treatment (since the injury happened at work), and occ med won't cover it unless you report it. But be ready for more blowback.
It is good that you are getting professional help for your depression, and I hope you keep up with whatever treatment has been prescribed. Also, as the needlestick is still on your mind, and probably will be until you do something about it, my view is that you should follow your institution's set procedures for reporting the needlestick, even if you are late in doing so (better late than never), and if they offer counselling, take it.
What does your residency contract say about taking time off for illness? Can whoever is treating you for your depression sign you off as ill, and what would the effect of that be? As you are in the last year of your training, it should be easier for your program to add in the missed time at the end of your year than it would be if you were PGY1 or 2.
I think you are right to be proactive in dealing with your worries about the upcoming ICU rotation - the worst thing for you to do would be to try to get through it when you are in no state of mind to do so, and coming up short. So I would suggest: firstly, talk to your medical practitioner about whether you could be signed off work, then discuss how you feel and what to do about it with someone sympathetic in your program (chief resident? resident adviser?). You will probably need to go to your PD, but it is a good idea to go with a plan: eg my medical practitioner is proposing to sign me off work for [x days/weeks], I've spoken to [chief resident/] about re-scheduling my rotations, I've spoken to [residency adviser?] about [eg making the time up later as specified in my contract].
Edit: I posted without seeing aPD's advice (a failure to refresh), and had no intention to try to trump what he said.
It sounds like you need a leave of absence ASAP, you need to get yourself right before you further damage your career. Make an appointment with the the head of GME asap. They can help you.
As for the needle stick, you're crazy for not getting tested and reporting this right away. You need to be tested immediately and put on whatever the current cocktail of meds is to try to avert disaster. Accidents happen, even to the most cautious, that's why there is a policy and plan in place for these injuries.
Whatever happens with your job, if you lose your health, you'll lose everything.
Good luck, and don't wait another day re the needle stick. Get on the meds and get the appropriate testing, today! I would go to the hospital's ED. That is where the after hours Occ Health stuff is handled at all my previous jobs/training. Seriously, today.
Don't risk your health worrying about more trouble for your waiting to report.
Go to your doctor and get a hcv ab and HIV checked. No point in getting yourself in more trouble if it's negative (and am extra week isn't going to make them more or less mad). From a medical standpoint, hcv is very much a curable disease now, so relax.
As for this rotation, tough call whether to try to make it through and then take a break or take a break now.