The Sandvich is incredibly useful, because it supplies you with full health and only takes 4 seconds to eat. Of course, 4 seconds is a long time in the middle of a fire fight. You can also use the sandvich to heal your medic if you press mouse 2 (By default). It will act as a medium health kit, healing half of your medic's (or anyone else, including enemies') health. If you have it out, and it is dropped (When you die), the ammo pack it drops will heal 50 health. Because that makes sense.
The actual sandvich will be able to be picked up by other heavies. You can get the sandvich by completing achievements for heavy, from a random drop, trading for it, or buying it from the Mann Co. Store. Or, you can buy strange/vintage/festive variants off the Community Market.

The Shotgun. Not much to say about it. It's a shotgun. It's useful because it shoots bullets that do about 80-90 damage a piece at close range, and you don't have to spin it down, unlike your minigun. This is the only real reason it's useful, other than if you run out of ammo. This is going to pair even better with the Brass Beast and Natascha, because these weapons take longer to put down than the others. It's going to be least useful with the Tomislav, which is put down faster. You start with the shotgun by default, however, there are other options as far as shotguns go. (Reskins, The Panic Attack, Etc.)

You may be wondering which to use, the shotgun or the sammy. The answer is, it's a matter of opinion. It all depends on your knowledge of the map. On Koth_Suijin for example, i like to hold down the control point when i play heavy. There are several health kits close to the point, but none of them are as fast as the sandvich. If you have the Gloves Of Running Urgently however, you will be able to get there faster. Two of the medium health kits are close to the spawns, so either way, there will be one near danger, which is bad at low health. On suijin, it's very easy to find a hiding spot. Under the many bridges, for example. Or, you can go right from the blue entrance to the CP and left from the red entrance to the CP to get to a bridge, and on this bridge you can get on the railing to jump up onto the roof, which is good for both surprising enemies and snacking on your sandvich. There is not one that is better than the other, however, a shotgun will be much more useful against classes like soldiers and pyros, which can do a large amount of damage in the time it takes to spin your gun down. In a highlander situation, where your medic will most likely remain healing you for the most part, i would definately recommend a sandvich, because 1. People in highlander are less insane then people in pubs, therefore less likely to be random places on the map, and 2. You WILL have a medic. A shotgun will also be useful against scouts, if you can aim it. Scouts can be hard to track with the minigun, so just flick your mouse around with the shotgun. Of course, shotguns do have to be reloaded, so you will definately need to watch out for this. Sometimes, it's better to run than fight, for example if a soldier or another heavy is on the point and you've got 20 health left. In these cases, it's usually quicker to eat a sandvich. So, in conclusion, if the enemy team is good, i would recommend getting a sandvich, just so you can heal up a bit quicker.