Is 不 A 不 B a grammatical structure like 又 A 又 B? I ran across 不知不觉 the other day.  If I use MDBG to look up c:不*不* I see there are many idioms with that format.  Can I use it with any verbs (?) or is it only part of fixed expressions?
Is 不 A 不 B a grammatical structure like 又 A 又 B?

They are different. Most [不 A 不 B] terms are either idiom or four characters idiomatic phrase that you have to memorize individually.
The A and B in [又 A 又 B] can be a single character word, a compound word or even an entire phrase. e.g. 又高又瘦，又高又瘦削，又想高大又想瘦削
Below are the common [不 A 不 B] terms:
[A complement B ]
不慌不忙: (not panic, not hurry) stay calm
不聞不問: (not listening, not inquire) ignore
不知不覺: (not know, not feel) not notice
不經不覺: (not experience, not feel) not notice (of time passing)
不瞅不睬: (not looking, not pay attention) ignore
不明不白: (not understandable, not clear) unclear
不倫不類: (not any type or kind) ridiculous
不慍不火: (not angry, not enrage) calm; mild
不痛不癢: (not hurt, not itchy) insignificant attack/ lost
不飲不食: (not eat, not drink) not eat nor drink
不眠不休: (not sleep, not rest) tirelessly;  relentlessly, persistently
不忠不孝: disloyal, not filial (no virtue)
不仁不義: unkind, unjust (no honor)
不過不失: (not considered a wrong doing, not considered a mistake) average/ mediocre
不折不扣: (no discount, no rebate) truly; definitely
不離不棄: (not leaving, not abandon) never abandon
不盡不實: (not detailed, not true) not telling the truth
不驕不躁: (not prideful, not irritated) stay calm and focused
不屈不撓: (not bending, not yielding) Never give up
不三不四: (not this, not that) dubious
不依不饒: (not agree with, not forgive) pestering/ not let things go
不聲不響: (Silent, not make a sound) silently/ be silent
[A contradict B ]
不卑不亢: (not too humble, not too excited) proper demeanor
不偏不倚: (not lend away , not lean onto) right on/ Objective (opinion)
不緩不迭: (not slow, not swift) steady normal pace
不男不女: (not a man, nor a woman) sex unknown
不上不下: (not up, not down) struct in the middle
不大不小: (not big, not small) the right size
不冷不熱: (not hot not cold) mild reception; right temperature
不高不低; (not high, not low) middle position (of rank or level)
不胖不瘦: (not fat, not thin) average body type
不咸不淡: (not salty not bland) unremarkable taste/ mediocre
[B until/ unless A]
不死不休: not stopping until one is dead
不吐不快: not happy until saying (something)
不時不食: do not eat unless it is in-season
不見不散: not leaving until we meet
I would update this list when I think up more.