That first syllable of besoin is a schwa, and hence is likely to be dropped:

J’ai besoin → jaibzoin

T’as besoin → tabzoin

Elle a besoin → (e)labzoin

That said, it will also be enunciated if someone asks « Répétez s’il vous plaît ? »

J’ai … be … soin. T’as compris ou ??

Similarly, for both "elle a" and "vous avez", yup, sometimes in rapid speech you might only hear the final (liaising) consonant of the subject. The first part may or may not be articulated, but it would be released too quickly or quietly to be salient. I suspect most of the time it’s just enough to distinguish elle/il and vous/nous, but see jlliagre’s comment that it can indeed be dropped entirely (again, I would be curious to distinguish perception from articulation, but it makes no functional difference).

It should be three syllables:
J’ai besoin: Jay – buh – zoin
T’as besoin: Tah – buh – zoin

And four syllables:
Elle a besoin: El – ah – buh – zoin

Yes, while technically there would be three syllables, we say it as if there are only two.

"J’ai besoin" becomes "jai-bsoin".

"T’as besoin" becomes "ta-bsoin"

This is however only in fast speech, usually amongst French speakers. If a French person is asked to enunciate or is speaking to someone who is less than fluent, they will usually say the full three syllables.

As for vous avez, this might also sound like two syllables in rapid speech:
"vous avez" then becomes "vsa-vez" or "sa-vez".

Again if they were asked to ennunciate they would say the full three syllables.