Does anyone knows if it's possible to ask for the supervisor to be excluded from the First Year Viva??

As far as I know the supervisor is a part of the Viva. At my university they have to sign a form saying they were present.

As for arranging the Viva, for me this was a case of harassing the supervisor, the internal reviewer and the post grad admin!

First Year Report & Viva - Feedback Form
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10. Viva feedback
Was the candidate able to explain the research problem and proposed contribution?
Was the verbal presentation of adequate?
Did the candidate understand questions put to him?
Was the candidate able to explain and defend his work to date and research plans?

11. Summary
Should the candidate pass the 1st-Year viva?
If yes, do you have any advice or comments for the candidate that might help with his/her future research?
If not – what course of action do you propose?
Is the supervision arrangement working? If not, what is needed?

Is the supervision arrangement working? If not, what is needed?

Regarding that question, if the relation between supervisor and student is not quite cordial and the supervision arrangement is NOT working then who can decide what is really needed?

Well, I think that if the relationship between student and supervisor doesn't work, this should be addressed before the interview. There is always a Head of PGs study, or if this role is covered by your supervisor, the Head of School to talk to. This is what I did at least. I didn't change supervisor, but things have improved greatly since then. Hiding under the sand is not going to sort out the situation. You could perhaps be a bit more specific and tell us what is your problem with him/her?

It depends on your department regulations. I had 2 advisors in a meeting not my supervisor, and it was not formally a viva. I told my advisors about problems.