A short fix is to put a large, washable rug (like from Ruggable) on top of the carpet where you want the table to go. That's not perfect as it will likely make it harder to pull chairs in and out.

If you own the home, a longer term fix is to replace the carpet with laminate that matches the kitchen floor (if you can find a match) and make it one cohesive space. If you decide to do that, you could put the table on the carpet for now and not worry about the mess since you'll replace the carpet. (I get that's not a great fix.)

If you want or need to keep the carpet as is, I think people will be able to help you better with a fully measured overhead 2D view of the kitchen, LR/DR, and adjoining spaces. There's an example in Kitchens Read Me First and I copied it below:

Congratulations for moving on with your life and getting away from the abuse.

There's gripping pads that you can put on your sofa feet so it does not move.

It's best to show more pics of your actual rooms rather than the 2D drawings.

More pics needed of kitchen to understand layout. In the one photo it looks like passthrough is on wall next to fridge - is that right? Why can’t the table be in front of the baseboard heating? It’s not going to block heat flow. Are you sure the closet in kitchen isn’t a pantry? Coat closet in kitchen seems odd & it looks like the door behind the doggie crate in last photo would be the coat closet. I agree that a table in kitchen makes much more sense for lots of reasons beyond messy eating with small kids. It would provide some work space & kids could use it for messy projects, coloring etc while you’re working in kitchen.

Yes, that's a passthrough next to the fridge leading into the living room/dining room. There is a small pantry in the kitchen next to the stove at the end of the baseboards in the photo. The coat closet is behind the doors in the photo with the dog crate, which is directly across from the stove, sink, and cabinets in the kitchen. I've read online that it isn't recommended to place any type of furniture near baseboard heaters due to risk of fire and I'm sure food/liquid would get spilled at some point and I don't want it getting on the heaters.

Ok - I’m still trying to understand how the 2 photos relate to each other - I thought photo with dog crate & stairs is LR with front door entry into townhome & not in same room as kitchen. I guessed the floorplan shown was the inside of kitchen only but hard to tell with what looks like baseboard heat running around most of room. But, here’s my understanding with baseboard heat & furniture. Upholstered furniture shouldn’t be smack up against the baseboard especially.That is a fire risk for the fabric. Wood or wood type furniture like a cabinet or bookcase etc with a long back might burn or scorch too & both the flat piece of wood & the upholstered back of a couch would block the heat from baseboard. Your question is about a kitchen table & chairs which are both open at the bottom so there is air flow & space. The legs won’t block the heat flow & won’t be smack up against the baseboard either. You have a corner space that’s possible for a table & chairs, yes? A round table would create more seating options. A table that’s extendable with a leaf would be a great thing for the times you really need a bigger table. The table needs to be far enough away from the corner to allow chair access to the 2 walls creating the corner. A pedestal table might be best so the chairs can be set at any position around the table with no legs blocking. I’m guess the baby will be sitting in a high chair for at least a year so for now, 4 chairs would work for you & the 3 older kids.

As for sofa sliding on floor with kids hopping on/off of it - that shouldn’t happen with the sofa sitting on carpet.

Hi, sorry it's so confusing. I couldn't really take an actual photo of the living room because it's packed full of boxes since unpacking has been pretty slow moving at this point. The floor plan is actually the living room. The actual photos are all of the kitchen. The entry door comes straight into the kitchen. Thanks for the info on the baseboard heaters. I guess I didn't really think about how a table wouldn't be flush against it.

Good for you! Once you are able to get the boxes sorted out and out of the living room, please give us pics of the room and a 2D drawing of the room with measurements. That will help a lot.