These words are all used in logics of various kinds, but they aren't all logical terms.
The first three are all about the same kind of idea, but expressed semantically, syntactically, and pragmatically. Only the semantics is truly logical.
Proposition is basic logic, as you say; Compound Proposition is an unimportant variant; it means it's a more complex proposition than usual, the same way
f(g(x)) is a compound function of
x. This is the semantic expression.
Statement is the linguistic equivalent of a proposition; it's one type of linguistic utterance, like Question or Order. It's not a logical term, because logic assumes that any declarative sentence can be either true or false, and that there are no other kinds of sentences. This isn't the case, but it's a useful fantasy in certain situations. This is the syntactic expression (in that it picks out one type of English syntax, and leaves out most of the rest).
Assertion is the speech act of uttering a statement with evidence of intent to convince the addressee that the speaker believes the statement to be true. It's a metalogical symbol, in that it can appear in syllogisms and pragmatic rules, but isn't part of propositional or predicate calculus. This is the pragmatic expression.
Argument, by contrast, is a term of logic (and mathematics), and doesn't refer to arguing.
Rather, an argument is part of a proposition in Predicate Calculus. Almost all predicates have arguments.
If the statement is
All men are mortal
then the complex proposition in logic is
(this would be read as "For every x such that Man of x, Mortal of x")
where x is the argument of the predicates
The proposition "Man of x" means "x is a man", and the other means "x is mortal".
Predicate adjectives and nouns are predicates and take arguments, just like verbs take subjects.
In fact, it makes sense to think of Subject and Object as arguments of the verb phrase.