BETO for Senator and (Vice) President!?

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Let's assume Beto O'Rourke wants to run against U.S. Senator John Cornyn in 2020, wins the Democratic senatorial primary and is later nominated as the Democratic vice-presidential candidate.

Is Beto allowed to run for Senator and (Vice) President at the same time?

If so, will there be a special election after Election Day in case Beto wins both the senatorial and the presidential elections, or will Governor Abbott appoint a new U.S. Senator?
If not, how will the new Democratic senatorial nominee be determined?

I know that some states allow candidates to run for two statewide offices at the same time. Joe Biden, for instance, won his senatorial election in 2008 and immediately resigned to become U.S. Vice President. Ted Kaufman was then appointed to continue Biden's term until Chris Coons was elected in a 2010 special election to finish Biden's term.
Are the rules in Texas the same as in Delaware?

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Lyndon Baines Johnson.

Interesting.
Johnson was allowed to run for both offices in 1960; he left the U.S. Senate on January 3, 1961.
William A. Blakley was then appointed to begin Johnson's term, but lost the special election in June 1961 against the 27 years younger Republican opponent John Tower. The question, however, is if the rules have changed in the meantime.

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If Beto wins the Senate election (which I consider unlikely), there would be a vacancy generated on January 3, 2021 which Governor Abbot would be able to appoint a replacement for until a special election later in 2021.

Beto could choose to serve in the Senate for 17 days between January 3 and January 20.

Could Abbott appoint anyone he wants, or must his nominee be approved by the state parliament?
If he indeed can, must the appointee be a Democrat?

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In the case of Bentsen in '88 and Biden in '08, they were entrenched incumbents with token opposition in their Senate races, and could afford to blow off Senate campaigning to run for VP, and still manage to win the Senate race.  In this hypothetical though, O'Rourke would be the *challenger* in the Senate race.  You really think he'd be competitive for Senate if he's not campaigning for Senate, but instead running around the country campaigning for VP?  I would imagine that for someone in his position, he'd have to pick one or the other, even though legally, sure, he could run for both at once.

I know it's just a hypothetical question. Maybe Cornyn doesn't want to run. Who knows?
But what would happen if Beto decides to step down as the senatorial candidate?
How would the replacing be done?

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I was wrong, Beto should be considered Harris Veep

What are you?

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Didn't Bentson do this?

And so did Biden in Delaware.
Rand Paul, on the other hand, wasn't allowed to do this in Kentucky.

Can anybody give me a list/map of all states that allows candidates both for president and for a second office?