First of all, I've taken carpet out of many areas where there were original hardwood floors underneath them. A hardwood floor is not structurally worse than plywood; in fact it is better quality wood and therefore structurally better. So, if I were going to do LVP throughout the home, and I was worried about the subfloor (the plywood part), I would pull the carpet up first to see what is under it. If there is structurally sound plywood under the carpet then you can make your decision from there about the hardwood. You could always raise the carpeted area to the level of the hardwood and do the LVP then, which could potentially save you money, depending on the size of each room.. Or, you can look for a reducer (strip that connects two different kinds of flooring), but that probably won't be your first preference because it will create a lip and looks old fashioned. Second, in my area (midwest) the market for good used carpeting and any hardwood flooring is so hot that pre-pandemic I advertised an entire second floor of carpeting, in good used condition (no stains, but used) on craigslist and within 24 hours it was gone. People came and took it all, padding and everything. Same with hardwood - family members took theirs out, replaced it with LVP, and sold the used hardwood at an overall profit. It is not hard to take carpet out; hardwood can be trickier depending on how it was put down. Seriously, I'm a 50 year old woman and with a claw hammer and some gloves I could get that carpet up in an 6 hour day, pad and all. Two preteens could take it up in half that time. Now if the subfloor is concrete (no basement), then you may have to scrape the glue off when you remove the padding, but even then glue is usually just in targeted lines, not evenly spread all over, so there is not much to get up.

Depends on your market/area/house. In our neighborhood/price range/house age, it would be out of place anywhere other than a finished basement and very much negatively affect the value if put in as the primary flooring, especially for appraisal. In other areas around here, it's right at home in the entire house and is getting put in a lot. On the other hand, right now you could sell a cardboard box anywhere here in a hot minute for over list price with 10 offers to pick from, so it may also depend on how far out you're looking and if this market continues for a while....

Remove the tile. It is $2/sf or so. Once gone, you NEVER have to deal with it again. Let's play the "What if?" game. What IF you have a problem with the vinyl AFTER it is installed. Let's PRETEND it is bouncy and squeaky and begins to unzip itself because of ALL the movement of the flooring. The FIX to the above problem (which MANY MANY MANY people have faced...just sayin') is to REMOVE the vinyl $2/sf. Then remove the tile $2/sf, then purchase MORE vinyl (you are guaranteed to lose as much as 25% of the vinyl with removal because the edges break...all the time). And then have it REINSTALLED for $3/sf. Wow. That's a LOT more money than what you would pay if you JUST REMOVED THE TILE! Nope. Not kidding. Oh...and don't let them fool you. The cost to 'raise' the subfloor where the carpet is = VERY expensive. The cost of plywood is NO LONGER $1/sf. It is NOW $3+ per square foot. If you have 500sf of carpeted area = $1500 worth of plywood. Now they have to INSTALL said plywood ($1-$2/sf) and then prepared the subfloor = $1/sf. Wow. Subfloor work JUST FOR the carpeted area = $4-$5/sf. If your quote is 'old' (more than 30 days old) you might just get a little surprise. Again, compare that to $2/sf for the removal of the tile. The REASON they do NOT want to pull the tile = TIME. They have you scheduled TIGHT against the next job. If you add in tile demo to the job, they are going to face DELAYS with SOMEONE ELSE'S job. And to me that sounds like a 'them problem' not a 'you problem'. Do the demo. I don't care how or by whom....just remove it. It SOLVES many problems. Leaving it in place CREATES problems. Just do the math. You will find the money isn't the issue. The TIMING is.

I don't have any experience with that specific floor, but I have sold dozens of Happy Floors tile floors and have never had an issue or claim with them. My in-laws put it in their 3 season porch. They love it and have not had any issues with it. About the only issues there are with porcelain floors are if the tiles arrive broken.