Q: What do I need to complete this heist solo without raising the alarm in almost any situation?
A: You need the following skills: Dominator, Smooth Talker, Joker, Cleaner (aced), ECM Specialist (aced).
It is much easier if you have more teammates because for example 2 jokers can convert 2 guards.
"Almost" any situation because, for example, if 3 Gensec guards spawn, it is impossible to complete it SOLO because there are too many guards. However, if you have teammates, you can take advantage of the joker skill and convert more than a guard to get up to 9 guards and remain in stealth!

Q: Why are you posting about the solo stealth?
A: Well, this isn't really just for solo. You can do it this way even when you have a full team.

Start by clearing up the street and the stores nearby. 3 civilians will walk in the street on Overkill. 2 on Normal, Hard, and Very Hard.
When you kill the civilians, make sure they are far enough from the bank and still in the map. You will need to put them in a bag and move the body somewhere well hidden.

After taking care of the civilians outside and the stores, go in the back of the bank and see if there are any cameras. Shooting cameras will make guards go check them. You can then take them out.

You can kill guards in 2 spots. The small room with the the red stairs (they will be facing the wall for a second. You need to shoot them in that second), or the other side (more dangerous because guards can see the body). I noticed that if you kill a guard in the room with the red stairs, most of the times a guard will not see the body even if they are there.

It is best to kill the first guard, then dominate the second guard to then convert them. It will require a lot of patience because you will have to wait for a few minutes.

After converting the second guard, place an ECM to delay the pagers and dectivate the cameras for a short period of time.
As soon as you place the ECM, go into the bank, dominate a guard, kill the other guard, shoot ALL the cameras and shout or kill the civilians. Lookout for the cameras, some are very well hidden. If you miss ONE camera, it is over.

After a few seconds the ECM will stop working. If you have done all the things mentioned above, you will have to answer only one pager. As soon as you answered it, you can do whatever you want until you place the keycards in the vault door. I usually take this time to move the bodies somewhere hidden.

It is a good idea to place boards during this time so that civilians will have to go in the bank to realize that it is being robbed and the GenSec guard will take more time too (thanks Mandala Barrier 'Yukari' [Mars] for this info). :)

Reminder: on Normal, Hard, and Very hard there are fewer guards than on OVERKILL.

After you place the keycards, a timer will start. The time is random. Sometimes it is only 2 minutes, sometimes it is 5 minutes. In this time, you will receive A LOT of phone calls. You will need to answer them.

Every time you answer to a phone call, some civilians will spawn (after a bunch of phone calls, they will stop spawning). Make the civilians go close enough to the bank before killing them. If you kill them too close to the end of the map, they can create a chain of alarmed civilians. Once there is an alarmed civilian outside the map, you have most likely failed the stealth even if you kill him (update: looks like it doesn't matter a lot if you kill them outside, but I would still be careful).

Lookout for the back of the bank! There is a chance that a guard will come out of the bathroom in the parking lot.

Sometimes there is going to be a guard coming from the bathrooms behind the bank..lol
The time is totally random, so you have to be careful. It is a good idea to keep 1 person looking the back if you're in a full team.
If you kill/dominate him and you are solo, you are okay as long as the GenSec doesn't come. If they do, you are f****d....yep...
If you are playing with other people, you should convert him, so you can use more pagers or dominate the GenSec in case they come!

After a few calls, Gensec will call. When Gensec calls, you are either going to get 2guards spawning, no guards at all, or going to search in the files to find the timerlock code. If you get the guard in the bank, the GenSec guards coming and you're playing solo, you have lost the stealth. If the bathroom guard doesn't spawn and 2 GenSec guards spawn, you can kill a guard, place an ECM while letting the pager start, kill the second guard, answer the pager, and then answer the second guard's pager after the ECM is done. Make sure you kill the guards from a good angle to not make them go outside the map.
You could also kill the dominated guard in the bank, then when GenSec comes in, you dominate one and kill the other. Your choice.

After that, the vault will most likely open. I recommend to open 1 to 2 deposit boxes at the time because if there is a phone call and the lasers block you, you will not be able to answer it and you will fail the heist.
To pass the laser, crouch, wait until the 2 lower lasers are off, and walk in.

There are going to be calls from the police captain. He will send 3 policemen the first time he calls. You can kill them without any problem. They don't have pager, but they can call the police, so kill them quickly (there shouldn't be a reason to hide their body because civilians will most likely have already stopped spawning).
The second time the captain calls, he will give you a few minutes before raising the alarm. There aren't going to be calls since then, so go into the vault and open as many deposit boxes as you can until you find the cash bag. Once you get one, you can escape!!

Thanks for checking out my guide!
Goodluck for the heist!
Please rate it so more people will be able to see it and learn about this! I hope this guide helped you! If you have any questions, please comment. :)