Does this also mean what would you like?
As in what they would like to eat/drink?Then yes.
wiktionary gives us लेएंगे for this form, i.e. with an extra ए. are both correct?
No. I think it's a typo. Repeating a vowel after its own mātrā like that looks very weird.
is it lenge plural because aap is like the highest formality
Yes. 'aap' conjugates with the plural form of verbs.
just to clarify. Aap conjugates to lenge, tum to logi and tu to lega?
Tum (masculine) would conjugate to loge. The rest are correct.
With feminine, it would be lengi, logi, and legi.