We selected a high end vinyl free floating wood plank flooring, for several reasons including patio / yard access direct through the 12 foot french door in the kitchen, sloppy dogs and general durability. Great wear and scratch warranty, 20 years. We are doing the entire main living area of DR, kitchen, den, foyer and powder room in the same. Installed over concrete and in the what used to be a sunken floor in the den, now plywood. Should be completed in a week, fingers crossed. We have friends who have a Karndean vinyl wood plank floor in their sun room. 11 years of no heat, soaking wet from rain and humidity and a fun filled patio and pool and beer pong marathons and it still looks like new. I was sold.

I have had wood look tile in my home for five years now. Went with Bedrosians Heathland Cherry Commercial Grade Porcelain tile. It has a hand scraped waviness to it so it is very "real" looking. The Commercial grade porcelain tile is heaver duty than a standard tile so will take more wear and tear than a standard tile. IT comes in many colors. https://www.bedrosians.com/en/product/list/porcelain/wood-look/ This company also has the tile that looks like marble too.

First off, check to see if you can purchase more of the engineered hardwood. If your floor is older than 2 years, you probably won't find the product ever again. Let's assume that's the case and move on to the rest of the question. Please DO NOT add wood-look tile to real wood. You can MIMIC the look of wood by using travertine-look porcelain tiles in the COLOUR you want. Striated travertine-look has the same 'grain' as wood. It comes in large format tiles (the same or LARGER than wood-look tiles) and it doesn't have to 'match' your wood it just has to compliment it. As for replacing the 'open' spaces with tile (including the front area to the right of the picture #2) it is your call. Right now it looks broken up. If you are looking to add visual flow with the flooring, then removing the engineered hardwood in that area will work very well. The next question becomes: what happens with the stairs if you remove the engineered wood from that space. Usually wood on stairs has something that it coordinates with. Right now that is the area to the right in pic#2. If you remove it, the stairs become a 'stand alone element'. This can be good or it can be very, very bad. To leave the stairs in place with the wood, you are FORCED to work with their colouring. As a stand alone element you have to WANT that colour in your house. And I mean DESPERATE to keep the colour. Does the wood continue UPSTAIRS?

OK...now to the MEAT of the issue. By 'rugs' I assume you mean wall-to-wall carpet. Ok. That means you have a house that has a second floor (can we assume wood joists for the second floor?) that has been BUILT to handle carpet. Carpets are light. They HIDE tremendous amounts of errors that the builders caused when building the house. At this point I would ASSUME the upstairs needs strengthening BEFORE tiles can be installed. Bathrooms are different. They have extra support because they are full of heavy things (tubs, showers, sinks, cabinetry). That means the bathroom floors are strong enough to handle tile. The rest of the upstairs is NOT strong enough for tile (if you have concrete then you can ignore this thread). Tiles are HEAVY. They will cause normal wooden subfloors to bend. Tiles don't bend...they crack. Their grout cracks. Then the pop up. Now you have a HUGE mess. The installer will have to do the math for the joists + subfloor thickness. That math will tell him/her how much THICKER the upstairs subfloor has to be. You will then pay for the new plywood for the upstairs. You will pay for that plywood to be installed (can be $2/sf for material + labour). Now you can install the tile. Tiles normally sit 3/4" thick once installed. That is VERY THICK compared to your carpet. Now the subfloor thickness is added to that. It is possible the upstairs floors will rise by 1" - 1.5". That means your stairs are going to get messed up. And now we have a BIG expensive problem. And that's just for the stairs. Imagine how big of a step down you will have into the bathroom. And the doors will have to be taken down and trimmed. If you want an easy to care for surface, you should look at laminate or vinyl plank flooring. Tiling the second story of a home can be VERY expensive. So expensive that most people do NOT try it. Be VERY aware of what you ask of your house. It was NOT built for tile upstairs. It was built for carpets...The stairs could be a nightmare you never ever want to experience. Their costs are extreme. Having to 'fix' a stair that is out of code can double, or even triple the cost of the project. No. I am not kidding. We've seen $15K spent on bring the stairs up to code in a situation like this. And then you have to pay to have a floor put on them.