Hey!
Okay, one thing after another:
- Typically, blood isn't used to produce vaccines. However, certain types of vaccines, especially protein vaccines, are produced in animal cells like eggs, yeast, or cell cultures created from fetal tissue. Generally, none of these are dangerous, if you're not for example allergic against egg. Even the fetal cell lines aren't actually a fetus anymore, they're just cells that grow and do their thing without ever forming a complex human being.
- No, vaccines generally don't cause diseases. You may have an adverse reaction due to an allergy or a very rare genetic defect, but both is very rare. However, there is a potential problem when there's a lack of needles! If the same needle is used to inject several patients, diseases may be transmitted from one to another. This is often a problem in third world countries.
- Vaccines generally don't do anything to your DNA. There are concerns that DNA and RNA vaccines may accidentally become part of the genome, but the likelihood of that is very, very low. So low that they're still considered safe.
I hope that clears some things up!