I took a look at your save and have a few ideas. I've got a new PC myself and the FPS was indeed single-digits, whereas my own fort is up near 100, so I see your problem. Here's my advice:
1) You have a huge number of items marked for dumping. Each item is its own job request. This causes the dwarves to path and sort this out, and can account for the "sudden" drop. If you don't find it too exploity, you can use DFHack to mass dump items instead.
2) You have a huge number of items, period. I saw over a thousand stone, cloth, food, plants, etc. - each! - in a couple cases two, all laying around. Tons of worn clothing. The biggest FPS gain I've ever experienced in any fort was after trading thousands of these items that I didn't need. I went from 1 FPS to 80 the second the caravan left the map. Make stuff out of the cloth, cook/mill/etc. the plants, trade the excessive meals and drinks (dwarves don't eat much), trade or smash the rest. I also suggest using quantum (one-tile) stockpiles with one-tile minecart dumps because big stockpiles seem to send out requests for items and lag everything to pieces. Let me know if you'd like an example on how to stockpile this way and I'll explain it in more detail.
3) You have a high population of dwarves and animals (200+ each). The animals run around and path in their pastures, too, so you could consider putting the non-grazers in cages or individual 1x1 pastures or otherwise limiting their ability to wander, and slaughtering any animals you don't need. I personally cage the babies, keep adults with "huge/fat" descriptions for breeding, and slaughter the rest.
4) Big designations like stockpiles, plant gathering/chopping, smoothing and so forth can cause sudden lag. Big open areas can do it, to a lesser extent, because units have to pick which way to go... it's usually a minor difference, but you could try using traffic designations with [d-o] to make preferred paths to get around in your fort.
EDIT: 5) I'm sure it's on purpose, but you have some dwarves locked in rooms with no path out, one of which is in a strange mood. Dwarves that can't path will keep trying, and that equals lag.
In my opinion a lot of the standard other answers aren't really a big deal in your case, if your PC is decent - your water is a brook, your embark isn't insanely big nor is your fort insanely old. There weren't many contaminants. You have some dfhack autoscripts running, which can slow things down a touch, but your PC should be able to handle it. I'd work on the things listed above - especially the second one! - and see if that helps.
tl;dr - you have too much stuff. Trade away your stuff.