Could I do Army ROTC, then do Infantry?

Yes, there are Infantry officers.
TECHNICALLY, 11B is for enlisted only and the code for officers is 11A, However, most Infantry officers and enlisted just refer to their job by the generic 11-Bravo title.

You don't do ROTC, then "join" the Army. In the last 2 years of ROTC, you will be a contracted cadet. In some limited respects, you are "in" the Army at this time. You will receive a branch and will be assigned to either active duty or reserve forces (Reserves or National Guard). At the moment you are commissioned, you will be a 2LT "in the Army". There is a short gray area until you report to your Basic Officer Leadership Course. You will receive orders for a starting date for the course. Between graduation and the start of BOLC, you can do pretty much whatever you want. When you start BOLC, you will be in uniform, going through the training to teach you how to be an officer in your specialty. This would be equivalent to AIT for enlisted Soldiers.

11B is an enlisted MOS.
Infantry officers are 11A.

officers do not have MOSs, officers have Branch's

so, no you cannot do ROTC and then be an 11b

11b's are enlisted

Yes, you can still be infantry, just as an officer instead of enlisted.

Officers don't actually get to pick their jobs though. You could just as easily end up a transportation officer and never work with the infantry at all.

You can request the job you want, but most people do not get their first choice. Some even get their very last choice.

Sure.

To elaborate on what Andrew said. You could be branched Infantry and never be assigned to an Infantry company, or even an Infantry Battalion. You could have assignments like aide to a General Officer, OIC of range control, an assignment in a higher level HQs G-3 department, could be assigned to TRADOC.