First, the symbol ⊂ is somewhat ambiguous because it may mean either proper or improper subset in different sources. To clear the ambiguity, ether use ⊆ for improper subset and ⊊ for proper subset or describe your notation in words.

An object x cannot belong to (S ⊂ T) because (S ⊂ T) is a proposition (something that is either true or false), not a set. It is sometimes possible to write x ∈ S ⊆ T as a contraction to "x ∈ SandS ⊆ T," but such shortcuts are better avoided in the beginning.

Only if x ∈ T, which, granted, is apparently the case here.

Or x ∉ T.

Strictly speaking, it just implies that S ∩ T ⊇ S. The other inclusion is obvious, but at this high level of proof detail maybe it should be said explicitly.