A few things ill chuck in there. Not 100 percent certain what your problem is but I have a few things to say which may be helpful.
1, stones do not go in bins. So making bins should have 0 effect on stone hauling. Stones are hauled to stone stockpiles by wheelbarrows, with up to 3 wheelbarrows assigned to the stockpiles. I would advise not assigning wheelbarrows to stockpiles other that stone ones.
2, bins can be a pain in the arse. Youve discovered that. They can be useful however. I use a strategy of employing bins in certain stockpiles and not in others. Leather, cloth and blocks I just leave the stockpiles with their max bin setting. Metal bars, armour, weapons, clothing I dont use bins. If you accidentally create a stockpile with bins on, the moment the game is unpaued jobs are generated to grab a bin and put it in the stockpile. The moment the job is generated you are going to end up with a bin full of stuff on a stockpile with a bin previously set to it. Emptying the bin is a pain but it can be done. (assign a dump over the bin,keystrokes - d, b, d then "undump" the bin on the stocks screen. If you have made a 1x1 garbage dump near to the bin the contents go into the dump but not the bin. The contents can then be claimed.
3,Your job cancellation spam I think is caused by the 1 guy with a bin who has "earmarked" his bin full of stuff. The people who would otherwise have picked up his earmarked stuff report job cancellation as they can no longer haul the item. This is like seeds in barrels. Due to a bug/issue with seeds currently, if you have seeds stored in a food pile with barrels set to it (and you want all food in barrels/pots) then 1 grower will grab the barrel and other growers will not be able to grab a seed from the barrel the 1 guy has picked up. Barrels hold a lot of seeds. Dont use barrels on seed only stockpiles.
4, If a bin has been assigned to a stockpile (like in the above description) and then bins are unassigned to the pile and the contents emptied. The bin seems to "hang around empty". It doesnt "collect" more stuff in the pile. Dump the bin and then claim it once its been dumped. I have seen issues where buckets get used to bring water to soldiers who are stationed. These buckets seem to get dropped and left where they are. Again I dump these and reclaim them. That clears up the buckets.
5, clothes get scattered everywhere by your dwarves over the course of years.Clothing wears out (x, xx, X, XX) and dwarves have a tendancy to drop their crap around. Also because clothes are "owned by dwarves" sometimes the crap lieing around cannot be hauled to the trade depot for example. Eventually x,xx,X,XX clothes do end up in the clothing pile eventually however given a single moment when you say "im going to clean up my fort" and dump a bunch of stuff, a lot of these clothes will "resist" the dumping because they are owned. If you use DF Hack this is what the clear owned command is for, which helps but I find doesnt eliminate the problem completely.
6 Im not sure why you have barrels and pots everywhere but again, making bins should have no effect on this as barrels and bins tend to be containers for exclusively different goods/stockpiles. Have you done much in the way of custom stockpiles ? If so delete these and remake from scratch with simpler settings to see if your problems are cleared.
7 Finally when using fastdwarf 2 1 (another df hack command) it seems that jobs are overlooked for watever reason. Dropping them to 0 0 will often get them to display on the units list a job they are now doing, that you want them to do that they were assigned to do but didnt before.
Destroying items is best done by an atom smasher. Build a 3 x 3 raising bridge. put a wall around the bridge and a single door in the wall. Link the bridge to a lever. Raise the bridge. Put a garbage dump under the raised bridge (i key followed by a gfor garbage zone assignment). When you dump items the items will be hauled to the garbage dump. So you can dump all your bins (make sure you have no stuff in them you want destroyed). Once you have your bins in the dump, lock the door, make sure there are no dwarves in the garbage compacting area, as they will be killed, then pull the lever. The bridge will lower "atomsmashing" anything underneath it. Your bins are now gone.
I wouldnt advise doing this with bins tho, just set all your stockpiles to not use em till you get the hang of it. They ARE useful and ive never had the problems you are that have been due to bins.
Good luck