Your first city in a region is likely most difficult to get started since you can't get any free help in the form of Fire, Police, Health and Garbage services from neighbors. You're going to have to build these items yourself.
To mitigate, give yourself a break on power generation. You first city should have abundant oil, coal or wind (and wind is a very distant third choice) to use in your primary respective power generator. This way you won't have to purchase power-generating raw materials on the global market, an expense which can quickly become a burden to a young city. If you're lucky enough to have quite a bit of your resources available, you can even sell the extra on the global market with a trade depot. Give up on the ideal of a clean power first city unless you want extra headache.
In addition, tourism and casino income is an easy first city moneymaker. Plop it early and add a police station nearby to handle any additional crime it generates. If you've got water or rail in your region, connect those up so you can get more tourists. Casinos are a great way to get a good deal of early money.
OK, since nobody has an answer, I tried it out myself in a private region.
Turning off addons did not affect the other city in the region, although it could be due to the region propagation delay. I'll check again tomorrow and see if anything's changed. I would imagine it's by design so one city shutting down or bulldozing everything doesn't wreak havoc on the region.
Also, only some buildings are shut down when I turn off the prereq. For the Dept of Safety, the Hospital will stay open while Fire and Police stations shut down.
For Dept of Tourism, Landmarks were still unlocked but big parks were not.
Looks like all the specialization unlocks stay after you close the HQs too.
Don't know if these are bugs or not, but right now you can save some $ per hour while keeping the bonus services.
Also, unlocks do not go across the entire region in the 16-city maps. They only go as far as the 4-city subregion.
Wish I could test this from sandbox mode, but everything is unlocked there from the get go.
Best Answer
No. City simulation is entirely client side as stated in this Rock Paper Shotgun interview.