Can I bring my own design?

Yes, you can. Almost every motive is possible but it will be converted into a tattoo design because some styles of artwork do not go well as a tattoo. Make sure the artist draws the motive again before he puts it on your skin so you know what it look like as a tattoo.

Can I be allergic to tattoo ink?

This is very rare.
About 95% of the people on this planet are not. The ink is always improved and they use only materials in it most people are not allergic to. I myself have never heard of a case like that, ever. But if you are still uncertain you could ask in a studio if they would be willing to "test" some ink on you. You will probably have to pay for it cause even for just one dot under your skin they have to go through the same sterilizing and safety procedures as if they where doing the actual tattoo.