The 7th attempt only worked because I got lucky and had many fewer than average murders (deadly fist-fights) and berserkers than on other attempts.
This was a very interesting challenge that's kept my attention for some time. I like jumping into challenge scenarios, because they give me a problem to confront without having to make mistakes in order to set them up in the first place. Trying to build a house while it's burning down around you is quite a challenge.
The challenges I faced and how I dealt with them:
1.) Dwarves running around chased by goblins.
Make use of the Burro(w) system in combination with Military Alerts. Set the underground and fortress interiors to be an area called "Safe" or something, create a military alert that has that Burrow selected, and then set that one as the "Civilian Alert". This means that all civilians will attempt to remain inside this area and not perform any jobs outside. It's the #1 most effective way of getting your people indoors.
You also have to re-open the inner gate to let them in. The double-layer of cage traps is enough to totally capture a goblin squad 90% of the time. Earlier attempts had..."incidents" where an solitary goblin warrior managed to make it into the fort and cause havoc (and death). So yeah, "safety burrows" are your friend. It's not an obvious thing to do, so it's understandable that you were at a loss as to how to save those who were trapped outdoors trying to retrieve the socks of their fallen comrades.
2.) No food or booze!!!
How did this happen? Seriously, your cupboard is bare. You have a ton of barrels lying around, so my guess is that you imported most of your food from caravans, had built up a large stockpile of food and drink, and then was in "Defecit spending" for some time until it all ran out.
The biggest challenge here was actually that tantruming dwarves will destroy farmland. I made extensive use of Dwarf Therapist to remove the "planter" job from anybody with a "miserable" mood, and set 100 tiles worth of farmland (in 1x10 strips to minimize crop losses) to produce plump helmets non-stop. The real saving factor though was that SEEDS ARE COOKABLE. It was "water and plump hempet spawn biscuits" for some time. I also trimmed down your large stockpiles to reveal open soil that periodically yielded shrubs that could be gathered. This gave some early booze.
3.) NO BEDS!!!!
Your population was what, 150, and you had a grand total of like 10 beds in your entire fort? Dwarves were falling asleep on the floor and having negative thoughts about "not having slept in a proper room". I built 3 carpenter shops next to the wood stockpile and had them produce nothing but beds. All wood was converted into beds (except for 5 stacks of wooden bolts for a marksdwarf security squad which proved invaluable in killing berserkers and cave toads, since the ammo bin had been dropped outside). While under a legitimate goblin siege (not just an ambush) in the summer, I was forced to strip out every log I could find in the entire fort. This involved deconstructing workshops that had been made with logs, and tearing up the wooden flooring that you used to seal off the cavern stairs, as well as tearing up some log staircases and a portion of your outer wall. All logs, and all destined to be beds (The z-stocks menu is invaluable for pinning down the location of stray resources). Beds are SRS BZNS. I built the beds in the engraved plus-shaped room that you had your forge in, since those would boost the room values. I built them 1 tile apart from each other with the bed zones being 3x3 causing only single tile edge-overlaps (which doesn't penalize room value much). I gave these "emergency rooms" to the most valuable dwarves to protect them from mood slippage, and as the bed quantity rose and the population fell, I was able to provide rooms for all. I'm still in the process of digging out proper quarters for everybody and moving people out of the emergency housing, but it worked in eliminating a source of unhappy thoughts.
4.) Barely any industry!
When you have a large population in an undeveloped fort, you can't get by with just "One of each kind" for your workshops. Trimming down your large wood and stone stockpiles, I made room for multiple workshops to be placed right next to their source materials for added efficiency. At one point I had 4 mason's shops carving coffins non-stop. I also enlarged the rooms in the "Dining hall level" and made a large and separated region for farms and food processing (so random tantrumers would kill tables instead of farms). I had 3 kitchens constantly turning out seed-biscuits to feed the hungry and have them leave the plump helmets alone for brewing. I also relocated the forge to next to your mega-stockpile. That level had the most free space of anywhere and already help most of the raw materials, so I ended up relocating 80% of all my industry to there. I also set up collection zones for fire clay and made lots of "Stoneware Statues" in the Kiln. Stoneware has a material value of 4 (compared chert's 1, so the statues were 100 apiece instead of 25), and is an easy "free material" that only costs the fuel that you had in such great plenty that you built walls out of it. I was baffled by this at first, but then realized that so long as there's no fuel shortage, why not use Coke bars as structural material. And there is no fuel shortage, and won't be for a very long time.
5.) Cut off from metal!
Tearing up the floorboards regained me access to the stairwell that led to the caverns, but more importantly, gave access to a vein of tetrahedrite. Copper ore! I mined that, smelted it up, and got axes and shields and light armor for a slightly pathetic melee squad. Due to interruptions by wandering naked mole dogs and giant cave toads (which helped the food supply when killed) I too sealed off the stairwell, but much lower down (z110) so I could keep access to the precious copper (and silver!) ore.
6.) Injuries!
Dwarves were getting bruised broken and battered from the incessant fistfights, so I set up a hospital zone so that a doctor could take care of them. Fortunately there was plenty of cloth available for bandages (and conversion into bags and ropes) splints and crutches, but thread had to be obtained by gathering wild pig-tails, otherwise no suturing could happen.
I also decided it would be best to have my population be dispersed so that fist-fights would be rarer. I thus eliminated all meeting areas and allowed the dwarves to roam free. I'm not certain if this was helpful or not happiness-wise (no social gatherings), but it did reduce property damage in the dining hall.
7.) Decoration.
Your dining hall was quite nice, and was a consistent source of happy thoughts for dwarves who ate there. there was a lack of tables though, which I had to fix. It seems that 1 chair per table is the way to go, either that or just more tables. At any rate lack of tables was a common complaint which I fixed. I also set up statues in the hallways so that they could be gazed upon, and I experimented with Sculpture Gardens (assigning them to the mayor when I didn't like the looks of people loitering in the newly created meeting area). Among my more ambitious projects though was to attempt to drain (tunnel to edge of map, then carve fortifications into the stone = infinite drain) the swimming pool which once would have served as a continuation of your central staircase. You pierced aquifer there, and so my drainage attempt was only marginally successful due to the constant refilling. It DID however create a pleasant mist which cleaned the dwarves and brightened their spirits...to the extent that an injured sober roomless dwarf who lost over half of their family can be made happier. At any rate, it allowed for the retrieval of Rith the Drowned Miner for a proper burial and retrieval of his pickaxe.
I also re-arranged and spruced up the quarters for the Mayor and Captain of the Guard so that they could both be happy. The fact that so many tiles were engraved helped with this immensly. I was able to get "Decent" quarters for both of them. This didn' stop the death of friends and family and the cares of the world from rendering them insane eventually, but I found suitable replacements. And then more replacement mayors when they either went miserable (thus unable to console) or were turned into statues.
When playing around with Sculpture gardens I found that each item placed within its confines (memorial halls from slabs work the same) generated a separate happy thought, so I went and just plopped a variety of different items in them to see if that would maximize their effectiveness. Don't know if it worked or not.
So yeah, that's kinda my story and maybe a bit of advice for you. My guess is that you got overwhelmed by waves of eager migrants without any clear plan for what to do with them or where to stuff them, and hence found yourself with a horribly overcrowded fort. It's easy to put off making beds and rooms for people, but it's really quite important. Sleeping in a good room is a steady supply of happiness. Oh, also, pasture your livestock in PROTECTED areas, otherwise the goblins will come and kill them all. But you already found this out, the hard way.
One last note about bridges as gates: Don't make them 1 tile wide! It makes it extremely difficult to tell whether they are open or shut, which is a really rather important thing to know. I ended up supplementing all your bridges with 2-wide partners behind them tied to the the same levers, levers that were walled off behind locked doors so tantruming dwarves wouldn't break them.
Thanks for the challenge!