If you're at all handy, you should be able to repair the mortar between the bricks yourself - any hardware store will have the materials, which you can mix in a bucket. If the structure isn't sound, though, it will be a continuous job to keep replacing the mortar. We had brick steps that were very similar to yours; we eventually replaced them with a wooden structure that has more of a "landing" where you can put down groceries etc while fishing around for the keys. It's not quite a deck, and it's not exactly beautiful, but it's handy and workable. The cost was much less than for a brick entrance, and although I'd eventually like to go back to brick, it's worked out well for us.

Lost me at #2

It's not clear from your post if the brick is entirely inside or a part of the exterior brick wall. In any case, just covering it up with sheetrock or anything else is likely to accelerate the deterioration. I would definitely consult a company or two or three that specializes in masonry restoration to see what they say. This is not a DIY project and is likely to be expensive, but if the brick is a part of the house's structural support system you really don't have many options.

Your pictures don't show much in the way of surroundings, so it's not possible to gauge what else the problem could possibly mean. But I doubt it is more than the common settling that is seen around foundations over the course of years. It looks like the steps have sunk more than the adjacent walk. The fix is either to have it jacked and support placed underneath (a process called mud-jacking) or to demolish it, compact the supporting soil well, and then rebuild a new steps, or steps and walk. Obviously, the latter option will be more expensive. However, it's a small job so might be relatively costly for the mud-jacking process. You just have to contact mudjackers and get prices to know where you stand on that. The mud referred to in mud-jacking is a concrete mortar mix. But a newer version of this process uses polyurethane foam. You might watch some Youtube videos of the process and see if it's something you could take a stab at (of course using home-spun solutions) as a DIY project. The jacking is done using conventional hydraulic and mechanical jacks and the mud or foam is squirted into the cavity created when the concrete is jacked back into position. A couple cans of Great Stuff might be enough to pull this off. If not, you'd be back at square one with little lost.