The catheter bag will be on a sort of stand or hanging on the side of the bed - either way you ought to be able to dangle a blanket strategically in front of it.
To answer some of your other questions - the catheter will most likely be put in by a nurse. Catheters come in different sizes - from really wee ones for babies to very large gauge ones that are used after men have had their prostates operated on. You won't need a large catheter, so you'll probably end up with one that is smaller in diameter than an ordinary pencil.
It's inserted into the urethra and passes into the bladder, and then sterile saline or water is used to inflate a balloon close to the tip of it, which rests where the urethra enters the bladder, so it doesn't fall out - you won't feel this balloon at all, and it is deflated again before the catheter is taken out.
A drain is used if a surgeon thinks that there may be oozing within a wound, and wants it drained out rather than forming a collection of blood, or haematoma that would delay healing and provide a possible site for infection.
I used to be an operating theatre nurse, which is how I know about drains and catheters, but I didn't work in the obstetric theatre (where c-sections are done), but as far as I remember from my spell on the postnatal ward during my nurse training, it is not the norm for someone to need wound drainage after a c-section - so don't worry about that.
I hope that helps you, PixieCake - and you are right to want to be well informed beforehand, as knowing what's happening to you helps reduce your stress levels, and that helps with your recovery.
Good luck and I'll be thinking of you and your gorgeous little baby!