I've never had any problems with different brands. I use BIC blades at the moment because you can do subscribe and save with them on Amazon, at least here in the UK. But you can sometimes get a better deal when buying in bulk.
I shaved my head with a safety razor when I first started as well as my face. I rarely cut myself on my face but it took a while to get used to holding the skin taut on my head to straighten it out before I stopped cutting myself there. But the cuts you do get tend to be very light rather than the deep gouges I got on the rare occasions my cartridge slipped off the handle. A piece of toilet paper should stop the bleeding and they should heal within a day at most.
Also, I'd recommend switching to using a shaving butter or soap rather than a foam. Makes it easier to see what you're doing.