What would happen if..

[True Federalist (진정한 연방 주의자)]

No, the Vice President does not get to break a tie in such a case.  The margin required for that vote is explicitly stated in the 12th amendment, "a majority of the whole number".  This isn't a vote where the outcome could depend on who shows up.  Say 51 Senators favor Adams while 49 favor Jefferson, but three of Adams supporters aren't there for some reason so that the vote is 49 for Jefferson and 48 for Adams.  Jefferson does not win despite having a majority of the 97 votes cast because it isn't a majority of the 100 that is the whole number.  The VP gets to break normal ties, but not in this case and not in Impeachment trials.  His sole role is to be the presiding officer in such cases (except when the President is being tried).

[True Federalist (진정한 연방 주의자)]

the bit about two-thirds being necessary for a quorum means that there have to be 67 Senators (2/3 of 100) present when they take the vote in order for it to count.  If they'd left out the bit about the wole number, it would only require a majority of those present (as is normally the case).  The Vice President is not considered to be a Senator, so I fail to see how he could be included in the whole number.  Given the absence of a tie-breaker when the House elects the President, I see reason to assume that they's have one for when the Senate elects the Veep.