Wow, what a terrible time of it you've had! Unfortunately, your type of experience is all too common amongst us TOS-ers. I've posted a lot of details on my surgery (which was done on July 25 this year) and recovery in the thread that was originally about Chris Carpenter (it's the one titled "Chris Carpenter St. Louis MLB, etc.), and another member is posting about his current post-op experience in that same thread. You may find some more information there helpful to your "length of time" and "level of care" questions.
I agree with NerPain that you will likely be unable to nurse or use pumped breast milk after surgery due to the drugs you will be given and the length of time you will have to be on them. You'll get opiate narcotics for sure and possibly a nerve drug as well (gabapentin, Lyrica, etc), in addition to whatever you're using now to control pain (i.e. flexeril, tramadol). In the hospital, you'll likely be given an injectable anesthetic (like propofol) and pain meds (like fentanyl or hydromorphone) before moving onto toradol injections and oral meds (like Narco). You also have to consider the possibility that after surgery, you could have a complication such as lung infection (requiring antibiotics, and sometimes steroids or other bronchodilators) or a blood clot (requiring blood thinners) - or other complication that makes it difficult to predict what kind of medication you might end up taking. If your TOS is not life-threatening and breast feeding is vital (i.e. your baby has a formula allergy or intolerance), you might consider postponing.
As far as holding your baby again, that varies from person to person, but you will be on a "nothing heavier than 5 lbs" restriction for at least a couple of weeks. Personally, I could have safely held a sleeping baby in my non-surgery arm with the support of my other arm at about 3 weeks post op, but I wouldn't have trusted myself to hold a wiggly, lively baby (i.e. one that might be able to squirm out of my arms) until after week 5.
As for your specific questions, I was able to shower (with no help) the day after I was released from the hospital. Ability-wise, you shouldn't have trouble pumping milk with a quality pump; I think the drugs would be your bigger limitation. You will want to sleep A LOT the first week - partly from your body needing to repair, and partly because good pain control drugs are sedating. By the time you are spending more time awake, you will be able to manage using the computer mouse, reading a book, doing crossword puzzles (or sudoku or whatever). I got a lot of reading done but was glad I'd stockpiled a bunch of TV shows on DVD to watch.
I was never actually in a sling at all (which I found very interesting at the time, seeing as how they'd put me in a sling after my scalene block - different doctors, though).
Good luck with everything to come - believe it or not, you've already done the hardest part - getting the right diagnosis.