Depends on the political jurisdiction and the politics of your area.
In our area, there has to be a public hearing and then a zoning board vote granting a variance before you can proceed. I looked into legalizing an illegally converted duplex to a triplex and the issues are:
1. Two families are built to a different standard to 3 and above which requires fire escapes, and cement enclosed furnace rooms, among other things.
2. If fire escapes are added, the objection at that point becomes one where it's a single and two family zone, and allowing addition of fire escapes to structures would alter the look and nature of the neighborhood, looking like a ghetto.
3. If you allow one guy to do this, you have to allow others, and traffic is already unbearable. I chatted with the head of the local home owners association who attends all the zoning meetings, he and his group shout down all such requests. His group is one main reason why a local car dealer cannot even convert an empty lot to a parking lot which he needs for his business, citing traffic. When I heard this, I said "wow", and the car dealer tried several times over a number of years, and in his last attempt, even invited neighbors to his business for coffee to ask for support. I was invited but the home owner association head was there too and convinced every one the answer should be no.
4. The schools are already overcrowded, where are we going to get funding for all the additional students and build additional facilities. This believe it or not is the one big issue at a town where I have a SFR, and I have to submit affidavits for tenants stating they are real tenants, and the students named is included in my lease, else the landlord is subject to heavy fines, reimbursing the schools tuition. Schools now operate in shifts, some students starting 7:00 AM and some leaving at 6:00 PM. So are we going for 24 hour schedules now?
After I looked into it, I didn't even try to get the variance in my case.
Each town is usually broken down into different zoning areas based on the master plan for the town. it all depends on what the area is zoned for, usually if it is zoned for single family residential you may be able to put a two family depending if that is a conditional use for single family homes in that zone. you can not just add on to a home and make it a 3 or 4 family without submitting plans to the town and going before the zoning board to see if they will allow the addition to be done. That said, i have been in the middle of purchasing a building recently, old school i want to convert to 17 apartments, it is in what they call residential medium zone which is up to 4 units, religious org, government use or a school. A condition of my purchase was that i would be approved by the town to build out the 17 units before i bought it. i went through the zoning hearing and was approved because they did not want to see the building just sit there and it was residential not some office building or stores and i was not looking to spot re zone the building. so in my case i was able to get the zoning variance but every case is different and you would need good examples and reasons why you should be approved.
To add to the above, additional issues that came up with adding new housing or added units to existing housing is the local utilizes, like electric, gas, water would chime in that the existing infrastructure cannot support additional housing. Where we are, water and sewer is operated by the city, so they are the ones to raise objections in this regard.
Everything you asked varies greatly from market to market and even sub-market to sub-market. To get answers on your specific area, try finding someone in your market and talk to them about zoning. They will be able to help a lot more than people in New York and Kentucky. Even better, call who ever is in charge of the zoning in your area and ask your questions, they may help.
this is literally the most street by street thing in ALL of real estate. Find a contractor who has built where you are eyeballing and ask them. Even if something is zoned R 2, 3, M.... local nimby's, affordable housing laws, and sundry other things can kill the project outright or make it cost prohibitive... In our area now, if you add on to a multi, you have to fire sprinkler that one AND the previous structure... $$$$$