As I understand it, the following is a principle (or "rule" if you wish) of English grammar:
Finite verbs take subject pronouns: I sing.
Non-finite verbs take object/possessive pronouns: That's me singing; She doesn't mind my singing.
Question 1. How do linguists account for the presence of a subject pronoun with a non-finite verb (i.e. That's how they found us; I staring out the window, Anne sitting by the computer)? Is "I" considered "non-standard" usage? Is it an "exception" to the rule described above? Or is the principle I mentioned inaccurate and therefore needs to be revised or discarded in favor of a more rigorous principle?
The principle above seems to work on contrast; that is, the form of the pronoun depends on the type of verb involved (finite or non-finite). That contrast doesn't apply to prepositions (as far as I can tell). In other words, it's not that some prepositions take subject pronoun, while others take object pronouns.
Question 2. what's the governing principle for the use of object pronouns in That's ok by me; Give it to him; It's up to her; etc.? I've seen some people talk about "Case" (i.e. prepositions assign object/accusative case), but doesn't "Case" imply/require inflection, and that doesn't really apply to English?
I suppose all of these can be explained by simply saying "that's how it is in English." I just wonder if linguists see an underlying principle at work for the above questions.
Thanks in advance