There (is, are) a light bulb and a screwdriver in the drawer.
We are asked to choose the article . The answer to it is are. I don't know why. I think the answer should be is, according to the grammar I have learned.
Whoever wins the bet should share the prize with the UsingEnglish users!
We deserve it !
About what you said - "THERE is not taken as the subject of the verb". I have been taught that 'There' in 'there is' is a kind of pronoun, much like 'dummy it' in "It is raining".
From Cambridge Grammar of English: "The so-called empty it and existential there do not refer to any object or entity. They are used as dummy subject forms (since a subject is required in non-imperative clauses) and refer generally to situations"
I can't say, grammatically, why I think it is singular as I don't have the grammatical knowledge. Perhaps because you can alter the sentence to say "There is a screwdriver in the drawer, and there is a lightbulb in the drawer." I have no idea.
Also, what you said -
"BUT if the first subject is singular and the second subject is plural, then most books approve a singular verb for the sake of sound: There is one light bulb and three screwdrivers"
- seems strange to me. For if both are singular (one lightbulb and one screwdriver) then it is plural (are), however if the first is singular and the second is plural (one lightbulb and three screwdrivers) then the verb is singular (is). With no real grammatical knowledge, this just seem irrational to me: if the rule for when one is plural is "take the first item, if it is singular then the verb is singular", then surely the fact that the first object in 'a screwdriver and a lightbulb' is singular makes this rule apply here too?
I genuinely have no idea, and am merely looking at it rationally. Although, grammar is hardly mathematical!