Learn Dutch: Heb je een zakdoek bij je? - Do you have a tissue with you?

Zich is only used for the third person (like hij, zij, or ze). For the first and second person, Dutch uses the regular object pronouns (me, je, ons) as reflexives. So, for 'you', it is bij je. If the sentence were about 'him', it would be Heeft hij een zakdoek bij zich?

Why is the preposition bij translated as 'with' here? Shouldn't 'with' be met?

When you are talking about having something physically on your person or in your pockets, Dutch uses bij. You use met when talking about accompaniment (like going somewhere with someone) or using a tool. 'To have something on/with you' is always iets bij zich hebben (or bij je in this case).

Could I say bij jou instead of bij je?

Yes, but it changes the emphasis. You would only use the stressed pronoun jou if you want to specifically highlight the person, such as asking 'Do you have a tissue? (Because I do not)'. For a neutral question, the unstressed je is the standard choice.

Why do we keep the preposition and pronoun separate instead of combining them into a word like erbij?