First I got stuck with the Pistol. Then I think I was able to Drop it and pick it up again. Then got stuck with the Prod. Not able to drop it. Mouse wheel (imprecise mechanism btw) nor keyboard buttons do not work to change the current weapon. Right-Click does not release the weapon either so I can't pick up anything.

I saw other forums mention how you have to equip, let it stick, drop/pickup, change to something else, let it stick, drop/pickup... a few times and it eventually goes away (???). This makes the first level nigh on impossible and if this bug continues it's not worth proceeding.

Anyone else encounter this or a cure?

Yes. If in single player, reload your most recent save from before the issue occurred. If in multiplayer, reconnect to the server. That's the simplest solution.

If you're looking for a more difficult one (IE the very one someone else was apparently telling you of) then I seem to recall the issue itself is sometimes caused by items in your inventory accidentally glitching and taking up the same space (usually via people trying to use that glitch to get more stuff in it) as other items; the game basically gets stuck with the current item out because despite it not being visible, two items or more are assigned to the same key, so it defaults to one of the items, usually the one in the top "layer".

You have to get all the items in your inventory either dropped, or reassigned to areas where they're not glitched on top of other items you can't see.

And again, this is a really weird glitch to even encounter; most people never do because they don't have enough items in inventory for it to even occur to begin with.

I'm encountering it now and moving stuff around as described above. It seems to help.