Strictly, in "proper" English, 'if' is only used in conditional sentences (e.g. 'I won't go IF it rains') and whether is used if there is no condition at all ('I don't know WHETHER I'll go'). In this sentence, therefore, whether is the strictly correct, formal translation, however, in everyday speech the words are interchangeable and in reality 'if' is dominant.
Yeah. We've a lot of uses for that style of contraction, and I know that the style "He's/she's/one's/you've/y'all've/they've/we've/it's/I've GOT X." is very common. Also, "He's made many (of) such errors." is perfectly fine as a sentence; I and many other speakers that I have heard, have used that structure before.
You can't say non so alone, it has to be followed by a subordinate : non so se... , non so quando... , non so dove... . To answer someone who asks you something, you have to say : non LO so. English, French, Spanish, even German and Chinese use the same "I don't know", not Italian...
Speaking as someone who has studied linguistics at university, linguists don't say "It can never be done". Linguists simply observe and analyze language as it is spoken. No judgements, no decrees of "ought/ought not". And it is quite common and valid to drop the subject pronoun in the situations you describe. It is a standard feature of the imperative to not use the subject pronoun, and it is quite common in everyday speech to omit the first person subject pronoun. (This is different from Italian's pronoun dropping/null subject. What happens in English is called "left-edge deletion" by linguists and does not work the same way.)
In Italian, verbs conjugate uniquely to the subject pronoun. Io, tu, lui/lei, noi, voi, and loro all have their own verb conjugations. So very often, it is not necessary to include the subject pronoun. The conjugation of the verb tells you who the subject is.
English uses do-support with its verbs for things like emphasis ("I'm telling you, I do know the answer!"), negation ("I do not think this is a good idea."), and questions ("Do you want to go to the movies?").
Italian does not do this. But like English, it does put the negation before the main part of the verb:
I do not know
The rest of the sentence is word-for-word how we would say it in English:
Non so se lui ha un cavallo.
(I do) not know if/whether he has a horse.