I don't know if you have heard of the lag problems with Dwarf Fortress, but anything approaching 200 dwarves will probably be completely unplayable. I suggest you set the cap to 130 if you want the full experience without 2-3 frames per second. Your dwarves will continue to reproduce after immigration anyway, bringing your total somewhere around 150-180.
That depends on your computer, really. I have... ~270 dwarves in my current fortress, and around 300 animals, and the FPS can still peak out around 20-30, lowering to around 10-15 when I order the exploratory mining expanded and they have to weave through my narrow mining tunnels, or when a few hundred new jobs are created to celebrate Goblin Christmas or whatever.
I should note that this fortress had a population cap of 200... but, well, I've been playing it a long time, and many, MANY children have been born, matured, and become productive new members of dwarven society. This has been the only death in the past few years...
So, on advice for dealing with massive fortresses... I don't really do anything special. Pretty much everyone has hauling on, but I disable hauling on dwarves that need to do specific tasks, and I primarily use Dwarf Therapist not to assign labors, but rather to monitor the progression of a dwarf's skills in the five invisible levels above "Legendary" (no way I was going to let a Legendary+1 Gem Cutter encrust my adamantine goblets with star rubies, only Legendary+5 will do!). Build lots of workshops, and use workshop profiling. Make huge stockpiles, make use of Z-levels to stack them on top of each other, don't make your average joe dwarf's bedroom too big if you utilize individual housing... a 2x3 or 2x3 bedroom is fine.
As for assigning dwarves to specific tasks... for those who work with basically infinite resources, like masons, stonecrafters, threshers, cooks, brewers, and clothsworkers (assuming you are growing the right plants), you can have a LOT of these and it won't hurt anything. Depending on your map, this may also include glassmakers if you have access to sand and magma. Training up engravers by smoothing out large sections of the fortress is good; I tend to not let them make actual engravings at all until they reach legendary skill through smoothing.
Miners do not skill up fast late in the game unless you do a LOT of mining, so I suggest training up a small mining force of 2-4 and going with that. Jewelers, metalsmiths, and for the most part carpenters and woodcrafters will have fewer materials available to skill up on, so I recommend keeping a smaller elite force dedicated to these jobs (then again, I aim to live at peace with the elves. If you would rather openly massacre the elves, you can probably go nuts with wood-based jobs). You can get by with only 1-2 legendary armorers, weaponsmiths, metalcrafters, and blacksmiths. It will be tiring to train them up the hard way unless you have magma and your map is very metal-rich, and even then you may not want to waste that much metal. Goblins only provide iron so fast. I have 4 magma forges myself, one is dedicated to creating armor, one weapons, one to blacksmithing, and one to metalcrafting, with only the appropriate legendaries allow to work them (workshop profiling).
Extremely niche tasks like lye making and potash making, if you do them at all, can be thrown on another dwarf and do not really require specialization. Remember, if you absolutely can't find a task for a dwarf, there's always the military or the guard.