I actually really like the rug! You have a lot of neutrals going on between the wall color, rug, and shades, but if you incorporate some color in other ways then your living room will be very warm and inviting. Colorful throw pillows, blankets, candles, art will all help.
I don't have a backsplash, so I don't have advice on that. I would prioritize your other purchases based on what you need right away. What is most important for you to have done? I only buy art that I really love, so I think you can get the essentials first and then be patient until you find the perfect art pieces.
I want to come back to this post later and offer up some suggestions. For right now, I definetly reccomend you checking out houzz.com for inspiration pictures. You can search for ideas using keywords and save the inspiration pictures you find to folders for rooms/concepts you are working on.
I have inspiration pictures on houzz that I based my decorating off of.
For example-
Inspiration pic:
My kitchen:
I liked the white cabinets, dark gray paint, subway tiles, and the idea of a chandelier that picture was my starting point and I searched other pictures for details and accessorizing ideas. You can create "idea books" on Houzz to save pics.
So I checked and I can link my Houzz ideabooks if you're interested
*poof*
Sometimes I like the whole room and sometimes I save a photo for one detail like the two-story curtains or flooring.
TBH I like white trim but I know the natural wood look is more popular up north. I just think white has a more clean look. I know H&G would not agree lol.
What are you planning on doing with the kitchen as far as cabinet/pull style and color? That would better help me find inspiration pics for you.