Hello, Chrissy! And welcome to someplace you would rather not be!!
The answer to your questions is "it depends." The purpose of the TURB is two-fold; one is to remove any visible tumors that are possible to remove, and the second is to determine the extent of the tumors and send them to pathology for a complete analysis. While your urologist is probably pretty sure that you have bladder cancer, exactly how far it has spread (if at all) and whether it if high grade or low grade will all be determined by the pathology exam. This information is necessary before a treatment plan can be set.
Your experience with the TURB will depend on where and how extensive the tumors are and this will vary from patient to patient. It is much more common for males to have catheters, but it may very well be that your urologist has noted where your tumor(s) are located and has decided that a catheter will be necessary. The "wash" with mitomycin is standard operating procedure to prevent any "loose" cells from the disturbance of the biopsy from re-seeding.
The questions you have about lifting your toddler are more appropriate for your urologist so you need to ask her what to expect. In my case, I didn't have a catheter and was back to normal the next day.
Wishing you all the best
Sara Anne