We just had our kitchen done. Order of operations was:

1. full tear out

2. electrical, plumbing, any window replacements, subfloor repair

3. wall repair

4. flooring of entire area. If you're putting in new cabinets it's better if the flooring runs under them

6. cabinets

7. counter

8. appliances

9. backsplash

10. paint

For some types of floor (such as tile), I would install the floor before the cabinets and do the entire floor (a little more expense).

Most DIY vinyl floors are floating click lock vinyl plank. If that is the case it should be installed after the cabinets and before the appliances. Installing cabinets on top of a floating floor is bad mistake. I would do all painting before installing LVP; all that should remain is hooking up appliances and baseboard.

Jennifer is right about the order. Before you start, make sure you have the ability to add the electrical outlets and update electrical in your panel. This is typically where a small remodel can start getting expensive. The plumbing should be pretty straightforward. I like to sub both of these trades out to licensed contractors because they can be problematic.

Hal is exactly right about the flooring. Know what your product is and how is should be installed. The manufacture has really clear specs about proper install. Typically you want the floor to cover the entire floor if you want to make changes in the future and are not able to find the same floor so make sure you save several pieces in case you have to do some repairs later on. Use a vinyl that has no VOCs. Typically inexpensive vinyls emit a small level of VOCs because of the plastics that are used. I would do everything you can to pay a little more for a better product.

I like to paint before any cabinets are installed. It is a lot easier painting when you only have to mask off the floors. One note about painting, Make sure the paint you purchase has no VOCs. For such a small space it will make a HUGE difference. Do your research about what is available. I would also ask find out about glues and finishes used in the cabinets you use.

I like using an induction cooktop, this type of cooktop will help cut down on heat in a small space, with a built in oven under it in the same cabinet in spaces like yours. This is a very easy way to help a small space feel less cramped.

There are some really nice washer/dryer combos that can help with space so you can tuck that into a cabinet.

If you reduced your dishwasher to an 18" model this will help you add another smaller cabinet which would be great for platters, cutting boards etc. These can be expensive but if you call around to speciality appliance dealers or show rooms, they are frequently changing displays so you can typically find one for the same price as a new 24".

Reduce the sink to a square single bowl deep sink which will help with adding another cabinet.

You have a great space if you spend some extra time designing it. This can be a great functional kitchen.

Good luck.