Getting a lot of pops & crackles

What condition do you think the stylus is in on a 30+ year old cartridge? I doubt it is good!

Would a worn stylus cause pops and crackles?
Also, the pops and crackles decrease appreciably if I play the same record again immediately. Maybe playing it the first time clears some of the dust. Is that possible?

No.
Yes.
What exactly are you doing, cleaning-wise, with your records, stylus, and platter?

Simply try another cartridge, you can invite someone with different cartridge if you don't have another cart at home ? Pretty easy

You can also play some new record or anything in MINT- condition to check

You'd better use this brush before you play any record.
Zerostat is also nice