Learn Farsi: باید کلاف را بجوشانید، به علاوه باید صبر کنید. - You must boil the skein; moreover, you must wait.

Yes, it is very common to say va be alâveh (and moreover). However, as this sentence shows, be alâveh works perfectly fine on its own to introduce the new clause.

Why is the verb bejushânid used instead of bejushid for 'boil'?

Jushidan is intransitive, meaning something boils on its own (like water). Since you are actively boiling an object (the skein), you must use the causative verb jushândan (to make something boil). Bejushânid is its present subjunctive form, which is required here because of the modal verb bâyad.

Is it grammatically required to repeat bâyad (must) in the second half of the sentence?

It is not strictly required; you could drop the second bâyad and just say be alâveh sabr konid, since the first modal can carry over. However, repeating the modal verb is very natural in Farsi and gives the second obligation just as much emphasis as the first.

Could I use alâveh bar instead of be alâveh in this sentence?

Not with this exact structure. Be alâveh means 'moreover' or 'furthermore' and connects two full independent clauses. Alâveh bar means 'in addition to' and must be followed by a noun or pronoun, so you would have to change the phrasing to something like alâveh bar in... (in addition to this...).