When giving blood, you could be experiencing vasovagal syncope -- fainting due to the drop in blood pressure and / or blood sugar. That's why they always hand out high-sugar juices -- hydration and sugar help reduce those effects. There are also sometimes symptoms before passing out. Here's a good article about it:

http://www.livestrong.com/article/166968-why-do-people-pass-...

However, that doesn't answer cuts. Does it happen on any sight of blood or only your blood?

I know a bit about this due to having researched after I gave blood the first time. Ends up the person was not very experienced and ended up drawing from muscle instead of a blood vessel. I was in for probably about 20 minutes with only half a bag before they gave up and sent me on my way. Walking from the parking lot to the office, I got lightheaded, nauseated, cold sweats, some gastrointestinal effects... But did not pass out. In my case, I think it's blood sugar related, as I had a similar reaction once while fasting.