It's too green, but all in all, it's not a bad kitchen, and if it is visible from your family room, it sounds like it can be made to work with your color scheme that you're planning for in there.
Paint the walls a golden yellow...not quite wheat, because that's too beige and boring, but like the color of a sunflower...maybe a shade lighter. Do that first. It may change your outlook on the floor and countertops.
If you still hate the floor after you paint the kitchen yellow, I'd do hardwood or some kind of laminate thing that looks like hardwood that is within your budget.
Get rid of that ceiling fan.
I wouldn't rip out those backsplash tiles...they aren't that bad and there aren't that many of them and with the yellow, they might "pop" nicely. Trying to repair a wall after removing tile is ghastly most of the time.
I like the idea of putting a new countertop on just your island and extending it to make it a breakfast bar, but if you can't afford to do that, I'd put wainscoting over the flat sides (any sides without drawers or cabinets) to spruce it up a bit. There are a lot of countertop materials that can be made to "blend" with what you have.
Really, all in all, I've seen worse kitchens. Heck, when we bought out house, my kitchen had white speckly formica counters and a backsplash wall covering made of the same formica in sheets, AVOCADO GREEN appliances, and a sheet vinyl floor that had flowers all over it in primary colors on a white background...think 1975. We did have nice oak cabinets though.