Both the it in (1) and the there in (2) are dummies; that is, they don't refer to anything,
and they're present in the sentence only to fill the subject slot.
the it in (1) is "Ambient it", a dummy it for deictic time and place
It's hot ~ It's noon ~ It will be too late then ~ It's raining.
Other dummy it types include "Distance" it, and the it inserted by rule in It-Cleft sentences:
Bill ordered salmon ~ It was Bill that ordered salmon ~ It was salmon that Bill ordered.
the there in (2) is the existential dummy there inserted by the rule called There-Insertion.
(examples in link)
So they're both dummies, of different sorts. As to whether they are subjects, ...
Subjects agree with verbs. Even when it's a dummy, it is still a third person singular pronoun.
It is raining is grammatical, but not *It are raining or *It am raining.
So dummy it -- of all types -- can be a subject. (There are also dummy it objects.)
Dummy there is a subject by some tests, and not (yet) by others.
For one thing, there isn't a noun, so it can't be either singular or plural.
Agreement with verbs in the list governing There-Insertion is officially
supposed to be with the displaced original subject, which can be singular or plural
There is a man here to see you ~ There are some people here to see you.
although contraction (and the difficulty of contracting there are) makes singular a default
There's a man here to see you ~ There's some people here to see you
There are other tests. But this is already too long.
Let it stand that one can consider dummy there to be the subject; or one can consider the displaced noun phrase after the verb to be the subject. Each is correct, for some value of "Subject".