How often should I wash my hands when cleaning the bathroom?
I generally wash my hands once at the end. I'm very conscious of keeping clean, but don't have any OC tendencies when it comes to germs. I'm not concerned about cross contamination while cleaning my bathroom. Ideally, the cleaning products are doing their job, and the amount of pathogenic germs around the sink, tub, and most of the floor are minimal. I'll use a sponge or scrub to wash the sink and bathtub first, then floor, and do the outside of the toilet and floor around the toilet last (I use a toilet brush for the inside). Doing it in that order prevents carrying any potential pathogens from the toilet to the sink where someone might set their toothbrush. I usually wear rubber gloves, but either way my hands are mostly just touching the sponge so they wouldn't be likely to cross contaminate anything.

I think in general people are a bit overly concerned about fecal contamination. A few weeks ago I caught my 1 year old daughter with cat poop in her mouth (gross!), I casually cleaned her hands with antibacterial wipes, and tried my best to wipe the inside of her mouth out. She is absolutely fine. Just an anecdote, but it at least proves that you're not guaranteed to have problems if you eat poop.

Just an anecdote, but hopefully it gives you peace of mind that you're doing a good job cleaning your bathroom and that you don't need to wash your hands multiple times throughout the process.