I'm a gentile and my name is Christian. Would I face more aversion if I wanted to covert to Judaism than someone whose name is, say, Abraham? Would I be encouraged to legally change my name? Also, there is no real significance to the name, my parents are not religious - my mom just liked the way it sounds.
In terms of rabbis being willing to work with you, I don't think that would be a factor. I've talked with a lot of converts and conversion candidates, including one named Christina, and none of them reported any inquiry or hesitation based on factors beyond their control like what their parents named them.
You will probably get some odd looks from other Jews and some may even say to you that that's not a Jewish name. You'll need to decide how to handle that; one response could be "maybe not usually". Christina told me she got fewer odd looks when she said her name was "Kris", for what that's worth. Or you could go by your middle name if that works better for you and you're uncomfortable with "Chris" or "Christian".
When you become Jewish you will be given (or will choose) a Hebrew name. This is the name that will be used for all ritual purposes, such as if you are called to the torah for an aliyah. Within your congregation you could tell people you prefer to be called that socially too; in shul you'll be Shmuel (or whoever) and outside you'll be Chris or Christian. (This is what Christina ended up doing.) If you can get used to being called two different things and occasionally having to sort out confusion when contexts overlap, you should be fine. Or you could go one step farther and use your new Hebrew name everywhere, either informally (like a nickname) or formally (legal name-change).
But to get back to your question: no, being named "Christian" will not handicap your conversion.
No, you would not have an issue because of your name. You may (depending on who you spoke with) get some desire to change the name, as some question the use of the word (although as far as I know it is better than substituting it with an X and writing X-tian as is common). But upon conversion the practice* is to take a new Jewish name, so for all matters Jewish it won't be relevant.
But think about how your mom would feel if you changed it, or went by a different name. It might be a reason to not convert.
*This is referenced somewhere on this site already, can't find where right now.
My name is Christopher (I go by the name of Chris). I am a Jew, and not a Jew from conversion, but born a Jew. I have met other Jews with the same name. My Hebrew name is Mashiach. Both names have the same meaning - of the Messiah (and NOT follower of Christ). Christoper, Christan, and Christina are all names that are perfectly acceptable as Jewish names. It's just ignorant to say otherwise. Although as a Jewish name it's becoming rare these days, however that was not always the case. In Sweden and Northern Germany is was a popular Jewish name before the turn of the century. Especially the name Christina for girls. The reason is historical. Queen Christiana (Christina) 1632 to 1654 of Sweden told the Pope to get lost when he tried to impose the inquisition in her kingdom. She defended the Jews in her kingdom and therefore they were never forced to convert or die like the Jews to the South were. The Pope branded her as a heretic but later rescinded it out of fear she would banish Catholicism from her kingdom and create her own church like Henry the 8th did. She became a hero to the Jews who named children (especially girls) after her for centuries to come. FYI, my paternal grandfather was Swedish. So yeah, Chris is a perfectly good Jewish name and one you should be proud to say is yours.