Wood – Filling in these gaps in door frame

Ok Kenny, here is the plan. You can remove one Jack stud and door header with a sawzall(cutting between the studs to cut the nails) and some work. You can then replace the jack stud with a 3/4" board cut to 3 1/2" width and the new 3/4" higher height. Put a 3/4" block on top of the remaining jack stud and reinstall the header 2X4. It will still fit between the outside studs and be supported 3/4" higher then before. Obviously, you will have to trim some drywall away to do this, but that can be pieced in and patched fairly easily and if you're careful cutting it before the demo begins. The drywall seam may be covered by the door trim. The result will be a finished opening 3/4 inches wider and taller than you started with. Your new door should slide right in! Good Luck.

Unfortunately, it looks like you have a structural problem in the front wall or foundation of the mudroom. This is evidenced by the severe cracking in the brick to the right of the door, as well as cracking in the concrete under your threshold and at left. The picture showing cracking to the right of your door also shows more mortar above the lowest course of bricks than in between the other courses; this suggests that this problem was patched before, but has recently gotten worse.

Considering that, it is likely that structural repairs will be needed. These could be incremental fixes, reinforcing some weak parts, or a more comprehensive fix (such as replacing the mud room's foundation or footings) could be required. Furthermore, if the failure occurred because of water/drainage problems, you will want to take corrective action there to ensure it doesn't happen again.

I would recommend you call some experienced masons and general builders to assess the situation. Because the mudroom is its own addition, the worst-case scenario is that the whole addition has to be rebuilt. More likely is that some significant foundation and masonry repairs are needed, somewhere in the $1000-3000 range. Once that's done you can do the door/carpentry fixes.

Note that this can be a complex issue to diagnose in person, let alone from pictures over the internet. You'll want to get several opinions from people who know what they're doing. A structural engineer may not be necessary, but their opinion could help you distinguish between the builders who understand what's going on and the ones who just want to sell you an overpriced patch.