Learn English – use meet for an online meeting

In your first example, you could very easily replace "Sure" with "Yes "and have very similar meanings. All it's trying to convey is an affirmative response, and I would not suggest that it carries and implied Please or Thank you. Think of it as short for the following:

Would you like a cup of water? - I surely would.

In your second example, the sure is definitely not trying to imply that the appreciation is expected or required. Rather it is one of several words/phrases that have come to be commonly used in reply to thanks. E.g. no problem, certainly, etc. These all have a similar usage in practice to You're welcome.

Finally, while this usage of sure is informal, it's common enough that I'd feel comfortable using it all but the most formal scenarios, with the caveat that in situations like your first example, I'd probably make it Sure, thanks since I'm being offered something.

Availability to meet, or availability for a meeting with me.

Though I'd go with your first choice: "Thanks for taking the time to meet with me". The other suggestions are, at best, excessively formal for most situations.