Let’s look closely at the three sentences that have been causing you such confusion:
1. “There’s only five classes on the second floor.”
2. “There’s many students not attending their classes.”
3. “There’s only two classes I got today.”
(Note than for consistency of style in the Forum, I spelled out the numerals in your sentences.)
Those three sentences are using what’s called in English grammar as the anticipatory “there” clause, meaning that the clause is introduced by the expletive “there.” In such sentence constructions, “there” is placed up front in the sentence or clause to point to the notional subject, usually a noun phrase of indefinite character, to give it more weight or emphasis. (Similar to this construction, of course, are sentences that use the anticipatory “it” clause, where the notional subject that follows is a nominal clause, as in “It is hard to justify his lackluster sales performance.”)
Now, the conventional rule for sentences with an anticipatory “there” clause is that the verb after the expletive “there” should be singular or plural depending on whether the notional subject that follows is singular or plural, as in “There’s a fly in my soup” and “There are three flies in my soup.” However, when the subject consists of two or more nouns compounded by the conjunction “and,” there are divergent schools of thought on whether “there is” or “there are” should be used.
Some grammarians prescribe that “there is” should be used when the subject consists of two or more indefinite nouns in the singular form, as in “There is apple and orange in our orchard back home”; to them, it just doesn’t sound right to say “There are apple and orange in our orchard back home.” Then other grammarians prescribe that “there is” can be used even for a compound subject for as long as the component nouns are notionally singular together, as in “There is fame and fortune for young people who can sing exceedingly well.” They argue that to use “there are” in such situations would yield such an awkward-sounding sentence, “There are fame and fortune for young people who can sing exceedingly well.” I think you can very well see and feel the validity of the point of those grammarians. Indeed, the problem with anticipatory “there” clauses is that they often put grammar and notion in irreconcilable conflict.
Now, in the light of the discussions above, are the three anticipatory sentences you presented grammatically correct and acceptable?
My answer is a categorical “no.” Those sentences are flawed both grammatically and notionally, so the correct thing to do is to replace their anticipatory “there is” with “there are.” As you pointed out, though, some native English speakers do speak that way—colloquially using the anticipatory “there is” regardless of whether what follows is a singular or plural subject or a compounded one. A conceptual justification offered for this tendency is that in day-to-day speech, the mind is normally not on the alert on whether the subject that will follow in a statement about to be uttered will be singular or not, so the singular “there is” can justifiably be used as the default anticipatory phrase. This can’t validly justify the resulting subject-verb disagreement, though, so I think such colloquialisms don’t deserve emulation whether by native or nonnative English speakers.
By the way, there’s a quick and handy way to avoid the grammatical conundrum in the faulty “there is” sentences you presented: knock off the anticipatory “there is” form altogether and rewrite the sentence with a suitable linking verb or active verb or both, as follows:
1. “Only five classes are held on the second floor.”
2. “Many students are not attending their classes.”
3. “I got only two classes today.”
These three alternatives are much more grammar-savvy than the colloquial “there is” sentences that have been bothering you. Your English definitely will be much better off with them.