Yes, don't worry about it .... When ever I clip my birds wings or nails they seem to feel "embarrassed" after. He/She will be okay in a few hours. A day or two at the most . By the way, I never clip to much. Just enough so the bird can't gain altitude but is still able to fly.:)

I have four birds

Yes, it's completely normal. Since budgies are flying animals, they rely on their wings! I personally detest how some people clip their birds' wings; it's not fair to the animal. If you've already done it, then I suggest just giving him some space to recover.

Yes. Budgies do not like being picked up and held. He is probably still shaken from the experience. Give him some time to settle down.