There are myriad ways in which you can phrase these and it would be grammatically correct, although awkward. Both of your examples are correct, but they would strike me as a rather clunky way to get your meaning across.
I would suggest the following, which is based on my own experience as a native speaker. It omits both "but" and "instead" and relies on the form for emphasis.
It's not your problem-- it's your friend's.
Depending on whether you want to emphasize the positive or negative aspect of the sentence, you can invert it, as in this example:
That's my toothbrush-- not yours.
In most cases, you can flip it either way without issue:
This will sound more natural and idiomatic to most speakers, at least in American English. I would love to know if other users have a different opinion.
Quick Note:
I've used dashes here to link the clauses of each sentence. This seems correct to me because the sentences have a corrective, interruptive quality. In my experience, native speakers are not always keen on the punctuation rules with commas (,), colons (:), semi-colons (;), and dashes (--). Disagreements over proper usage can sometimes create opposing camps (e.g. the Oxford comma).
For what it's worth, I would say that you could substitute each dash (--) with a semi-colon (;) and the meaning would be largely the same, if a bit less emphatic.
No, your such people like structure is not correct. It might be clearer to you if we arrange it in a way that's more natural to most modern English speakers, with such + as together:
He tends to easily get close to people such as Tom. ← Correct
He tends to easily get close to people such like Tom. ← Incorrect
That combination, such like, is only ever heard in very limited regional varieties of British English, in the phrase "and such like", meaning "and other things like that", and should really never be used by an English language learner.
Best Answer
All of the sentences are grammatically OK. Consistency of tense is more about good style than proper grammar. However, be aware that it's also good style to omit repetitive words when they are not necessary, as in your first sentence:
The second "he should" is not necessary, since the reader already understands that the writer is offering some kind of advice. So these should both be read as the same tense.
The other sentences are not as well written, but still natural English. More parallel comparisons could be written:
Unfortunately, good English style is difficult to teach. The best way to learn is to read a lot, and emulate the style of good English writers. Most newspapers and other large periodicals have good writing style, but even this varies between something like The New York Times or The Wall Street Journal, and some smaller newspaper, or between Time Magazine and People Magazine.
[Edit] As virolino mentions in the comments below, repetition is not always bad. It's fine when it serves a purpose, such as to emphasize or clarify. You have to judge from the context whether it adds to or detracts from the intended meaning.