In English, when you have a compound subject with "and" or "or", the rule is: If it's "and", the subject is plural. If it's "or", then if each part is singular the combined phrase is singular, and if each part is plural the combined phrase is plural. If one is singular and the other is plural, you go by the one that is closest to the verb in word order.
So, I think a declarative sentence might be easier to begin with ...
Al and Bob are ...
Al or Bob is ...
The dogs or the cats are ...
The dog or the cat is ...
The dog or the cats are ...
It's the same rule with a question. Less intuitively obvious to me, I have to think about it sometimes, but maybe that's just me ...
Do Al and Bob ...
Does Al or Bob ...
Do the dogs or the cats ...
Etc.
I see that someone in the comments noted that "and" can sometimes be a part of a name or identifier of something, like "Ben and Jerry's". In that case the full name would be "Ben and Jerry's Ice Cream Parlor" (or something like that, I don't know their official name), so that's singular: "The Ice Cream Parlor is ..." "Ben and Jerry's" is an adjective phrase modifying parlor, and doesn't change "parlor" from singular to plural.
It might be a little more confusing if we were referencing something where there is no singular word in the full name. Like, "'The Avengers' is a TV show starring Patrick McNee." (Sorry if the pop culture reference is way outdated but it's the first one that came to mind.) "Avengers" is a plural word but it's the name of a TV show, which is a singular thing.
So short answer: "Do this and this ...", not "Does". There are two things, the first this and the second this. Assuming that "This and This" is not the name of some single thing, which in this context, I think you're saying they're two web sites.