Most BB creams do have SPF, but it's not nearly ENOUGH SPF unless you're wearing a ridiculously heavy layer of it. Wear sunscreen underneath your BB cream.
You need to reapply sunscreen after two hours of sun exposure, not two hours of wearing it. If you're going to be in the sun all day, maybe it's better to skip the BB cream. But if you're just moving around the city and going to work indoors, you shouldn't actually need to reapply sunscreen.
Depending on exactly how pale you are, most Korean BB creams should work for you. They tend to run pale. Unfortunately, I have very dry skin, so I can't help you so much with a brand recommendation. I know Dr. Jart is a favorite of many people, and you may be able to get it at your local Sephora. That's a good idea, because the sales associates can help you find the right color. And don't be embarrassed about it - they're used to seeing dudes in there looking for tinted moisturizers etc. Very normal thing.
As for patch testing, you'll want to do it the same way as for most skincare. It's unusual for a BB or tinted moisturizer to cause a problem, but try a little on the inside of your wrist first. If it doesn't bug you after a couple of hours, you're good to go with trying it on your face.