Can someone living with a non-Jewish friend/house mate convert to Judaism? Obviously, there is a concern over issues such as Shabbat and Kashrut - however if the non-Jewish person complies or makes compromises would the Beth Din allow the person to convert? The flatmate is vegetarian so that's one less thing to worry about. We will however have separate dishes - my sets for meat, dairy and pareve, and theirs.
Monica beat me to it by a few seconds here, but I'll finish what I started writing.
I'm assuming the housemate is not a "boyfriend/girlfriend" type person. Just someone who happens to live in the same house. Otherwise, that's a horse of a completely different color!
(You may want to rephrase the top question line, as sadly I think many people today would read "living with a non-Jew" as "having a non-Jewish significant other.")
Basically, a panel overseeing a conversion is asking a few questions: "Is this person sincerely trying to keep all of Jewish law? And is s/he on a trajectory where s/he remains most likely to do so?" The particulars of how to determine that will vary from beit din to beit din.
It would not surprise me if some rabbis felt that having a non-Jewish housemate makes keeping Judaism difficult enough that they wouldn't endorse it. (Notice how many observant Jews seek out other observant Jews as housemates.) Otherwise, in a theoretical sense it's doable, but there are a lot of questions to ask (this comes up with observant students looking into college dorm options as well), such as:
I don't think it's inherently a "red line", there are enough little things that can go wrong if you don't really know your stuff. Some rabbis may say "okay we'll work it out"; some might recommend trying to find a Jewish housemate if possible (which will probably create its own bunch of questions as your learning mistakes will make it difficult for your housemate); and others may say "it's just too much of a mess."
This is something to discuss with your rabbi, but I'm not aware of any halacha that would forbid this and I've never heard a convert (I know several and we've talked about the process) mention this issue. That's weak evidence, but consider: your rabbi (and ultimately the beit din you stand before) is concerned that you will be able to transition into and maintain a proper Jewish life. If you can do that in your current living situation, then there is less need to ask you to make changes.
However, you will need to have detailed conversations with your friend/housemate to make sure the implications are really understood, and then try it for a while and see if it's working. It's one thing to agree in principle and another to be confronted with the details all the time, like separate dish-washing sponges, lights being left on that one instinctively wants to turn off, and so on. Everything that you are working to re-wire in your brain as you make the transition will be ten times harder for someone else, who is not motivated like you are, to do.
All of this is for a housemate who is the same gender as you and not a romantic partner. For those cases there are other issues that you definitely need to discuss with your rabbi.
A note on one detail you mentioned: if your housemate is vegetarian but not vegan, there may still be kashrut issues with the shared kitchen appliances (vegetarian includes dairy). Are you prepared to re-kasher the oven every time you want to cook meat, for instance? You will find it far easier, if the housemate is agreeable, to just maintain a kosher kitchen at home.