Dying your hair while pregnant is not all that dangerous. =] Read up on it. Many, many health professionals have stated that it's alright to do it - but if you're still "wary" of it, to just not do it during your first trimester. Their largest concern, is the fumes you breathe in. Having your hair done in a well-ventilated place reduces this significantly.
That being said - tattoos - no fucking way. And any REPUTABLE tattoo artist, would refuse the service to you, as is their right. My tattoo places have signs posted right on the wall that say they won't tattoo if you're pregnant.
Though, I agree with Greta. Any tattoo artist should be opening the package of needles in front of you, and should be disposing of it properly in front of you, too. The place I go to makes you watch them bend the needle in half (it can't be used after that, won't go in the machine right), and dispose of it. It really shouldn't be that hard to find one that does it right, Mistress Snape (why did I feel naughty typing your name? LOL).