So, as Democrats in the Senate lick their chops, now that they will officially control the Senate after inauguration [because until inauguration, Kammie is not the tie-breaker] over a trial on impeachment...
Wait, that may not happen at all, because in spite of an ill-advised second House impeachment of Trump, it must be decided, by interpretation of the Senates power to try an impeachment after the President has left office - and the Constitution is fairly clear, since "removal from office" would not be a consequence as of noon tomorrow...
... is disqualification an automatic? No, the Senate votes on that issue separately. Are both issues automatic, i.e., removal and disqualification from holder future office? No, because of the last answer. Can the Senate not vote to remove, but still vote to disqualify, i.e., remove, or disqualify? No. The operative word is "and," not "and/or," or simply "or," therefore, if the Senate takes up the trial, their must be a conviction/acquittal vote, and only then would the Senate decide to vote on disqualification. That is, unless you want to treat the Constitution like a cafeteria, in which the words can be altered at will. But, that's how Democrats read the Constitution, anyway.