According to Conservative Judaism, if a Kohan is married to a Conservative convert, does he lose the ability to perform the rituals of a kohan-ie the Kohan Alya etc.?

Shalom, Thank you for your question. The truth is that as an Orthodox Rabbi I am truly unaware of what the Conservative Judaism movement says about a cohen marrying a convert – so I can't help you there. I can say that from an orthodox position a Cohen it is forbidden to marry a convert. (There are also issues as to the validity of a conservative conversion altogether). If he none the less did so, there is a ban on giving him any honors – such as letting him recite the cohanic blessing, or receiving the first Aliyah to the Torah. The application of this ban differs from community to community, with some being more strict or lenient than others, but in general such a cohen may not be honored by the community. I hope this is of some help to you. Blessings.