Disclaimer: I'm not an airline pilot so don't just take my word and apply it in a real world environment
From my understanding:
TOGA means takeoff/go around and is usually a thrust setting of the engine. It can also be the vertical or lateral mode of an autopilot/flight director. I haven't seen anyone referring TOGA as a situation of flight though. "I've flown a TOGA" is a thing that I have never heard.
Touch-and-Go: Performing a landing and aborting it after you have touched down. Usually this is planned ahead of time and ATC issues a "cleared touch and go" or "cleared for the option" instead of "cleared to land". Usually ATC doesn't expect a message from you during the go-around phase as far as I know.
Go-Around: Aborting a landing in general. My understanding of a go-around is aborting an approach before touching down. ATC usually expects a "going around" message from you.
Aborted landing: Similar to go around but my understanding would be that it takes place after you have touched down. The reasons may be a bounced landing, a late touch down with too little runway remaining or things of that nature. They are unplanned in comparison to a planned touch-and-go. ATC probably expects a "going around" from you.
Stop and go: Performing a full stop landing and taking off right after. Should be requested from ATC and depending on traffic it may not be possible. It gives you time to go through all checklists before taking off again and does not require a long taxi back. Usually this is only done by general aviation aircraft when the runway is long enough.
Now to the "Can You Do It?" question:
With a long enough runway you can easily perform a full stop landing followed by a normal takeoff ("stop and go") or a "touch and go". The worst case is an engine failure at V1 during the 'go'-phase. You'd have to guarantee that the aircraft will be able to stop on the remaining runway or that you are above minimum control speed in order to stay on the runway and climb out with one engine inoperative. This is probably the reason why it's usually not considered as a safe option to go around after you have already slowed down significantly.
There are many other factors that have to be taken into account though:
- Operating manual may prohibit it.
- Company rules and guidelines may prohibit this even if may be physically possible and safe to do so.
Actions that will have to be performed during the touch and go:
- Ground spoilers should retract as you move thrust levers forward. You have to verify that this happens.
- Autobrakes will disable almost instantly once you move the thrust levers forward. You have to monitor that.
- The feet of the other pilot in the flight deck have to be off the brake pedals. In an emergency situation it could happen that the rules of "pilot flying" and PNF are not respected by the other crew member and that they think the autobrake may have failed and they try to stop the plane. Communication needs to be clear.
- Pitch trim has to be reset for takeoff. You have to do that manually in most aircraft
- Engines have to be stabilized for a brief moment before adding full thrust if you are slower than minimum control speed.
- Time between commanding full thrust and actual engine thrust can be 5 seconds or longer.
- If you applied reverse thrust and switch directly to forward thrust you will need to make sure that both reversers are stored and properly latched closed before you can apply lots of forward thrust. The last thing you want is both engines on max thrust with one reverser still open.
- Brake temperature. If you have already decelerated significantly then your brakes will be quite hot but the indicated temperature usually lags behind. Max indicated temperature is usually reached several minutes after landing... So you don't actually know how hot your brakes are until minutes later. Before taking off again you should verify that brake temperature is not too high and within limits for a potential rejected takeoff. If you decelerated only a little bit and noticed that braking action is poor then there is no physical reason as to why the brakes should be too hot.
- Flaps will need to be raised from full landing configuration to go around position. It may take several seconds before the flaps reach that position. During this period you have a lot more drag which reduces the acceleration.
During all this time you have you're just wasting runway length. If you're going 130 kts for just 5 seconds this means you waste roughly 330 meters or 1100 feet before you start accelerating again.
- Due to the retraction of the flaps you reduce drag but you also lose lift. I see no physical problem doing that while you are on the ground rolling but if you are already airborne again you may need to wait for a good positive rate and good terrain clearance before you can attempt that.
- Climb performance with flaps extended may be significantly worse - so terrain and obstacle clearance have to be checked.
Usually there won't be time to check the proper pitch trim setting, check the proper V1, VR and V2 speeds during such an event. And if you have enough runway left to do all that you may as well stay on the brakes and add full reverse thrust to stop.