The idea of "house signature" really varies from maker to maker. Some makers are quite good at hitting a house target - Hifiman for instance has consistent frequency response in a lot of their headphones - but some manufacturers are all over the place, i.e. Stax. So really it's a case by case basis.
There is some correlation between price and performance, but not that much. Realistically, you can get good sound at (nearly) every pricepoint, depending on what you define as good of course. Diminishing returns hit hard however, and above $300 or so you really see much less linear gain even if you were to only look at standout products.
There are some exceptions - some electrostatics for instance essentially break diminishing returns giving you vastly better performance compared to cheaper models - but overall diminishing returns hit hard, and honestly the answer to what you get when you go to $1000... not nearly as much as you get going from $50 to $300. Mostly, headphones get more resolving, maybe a bit wider staging, a bit better dynamics and separation, but critically the tuning doesn't really get any better. This is why learning to EQ properly can get you better results than simply spending more.
The very best systems can be pretty special though. Good headphones can trick you into thinking that there are instruments spontaneously appearing out of the air all around you rather than a pair of drivers on the sides of your head. But I've only gotten that effect a handful of times - the HE90/HEV90 combo did that as did my old SR-007/Blue Hawaii rig, and a few others got close (L700 does it but not as well, HD650 sometimes off a good tube amp, Utopia or Clear with DSHA-3F kinda sorta, etc). These systems can sound special but building one takes effort and time, and you cannot force it by simply throwing money at the problem. Not everything expensive is good, or wants to work together well.
Lastly, when you're spending that much you always have to look at what else you can get at that money. There are some amazing nearfield monitors you can get or even bookshelf and floorstanding speakers at the higher pricepoints, and they will do things that no headphone system ever can, so you always have to pause and ask yourself if it's worth it.
Open back vs. closed, well I don't listen to much closed so can't tell you there.
Music has an enormous impact. The things that you index for in your preferences depend greatly on what you listen to. If you listen to a lot of vocals, acoustic, jazz, etc then tonality and tembre are very important, as well as midrange quality and performance. If it's a lot of electronic and more dance oriented stuff and you're used to listening to it on club systems, you're more likely to index for strong bass. I'm not gonna go through the list but it matters a lot.
TL;DR try to listen to this stuff yourself, maybe find a brick and mortar store or go to a meet, that'll answer some questions. Ultimately in anything that depends on perception, there will be a great deal of different takes and disagreement, and there really is no substitute for your own experience. And of course your own preferences will evolve over time. You might find something that shocks you or sounds really unique and start wanting that, only to get tired of it and start indexing for something else years later. So there's really no amount of posts you can read that will tell you all this stuff, you just have to take the plunge and do it yourself.
Taking the time to understand the nitty gritty, learning to read graphs, learning some actual sound science (there's a lot of bs masquerading as science in audio) will speed up the process and will help steer you better towards the stuff that's legit, but of course learning this takes effort and time too.
Personally, I index for effortless resolution and accurate tonality above all else, and I also EQ. For critical listening I've been using electrostatics for decades since there is nothing else that resolves like them while at the same time sounding smooth and natural. However there are lots of fairly crap sounding electrostatics out there today and my favorite systems are mostly the standout older ones. Generally when people talk about electrostatics sounding ethereal, diffuse, too dry, having no bass, etc - the bad ones sound like that, the good ones sound natural, very liquid and smooth, are pretty well tuned, and where they're lacking EQ works wonders.
On the flip side, for casual and all-around listening, I have yet to find something that works better than the HD650 off a good tube or hybrid amp, it's very light, comfortable, nearly perfectly tuned, not particularly picky about source material while at the same time being resolving enough to hear just about everything, and I can listen to it 8 hours a day and not notice. As you get older, stuff like that matters more, so I end up using the HD650 95% of the time even when I have some much higher-end headphones lying around. But it has to be in the right rig, with your average solid state amp it can sound woolly or dull and not have any of the magic it can have in the right system.
All this stuff I found mainly through trial and error, trying and rejecting things I didn't like to eventually find the stuff I did.
This leads me to the last, and most important point:
Trying and rejecting things you don't like takes up most of your time, and seems like a waste of time, but it isn't. 95% of the stuff out there won't work for you, but you need to go through it to find the 5% that does. This is true for any hobby, any skill, anything you learn: most of what you learn will be useless, but you have to go through it all to find what's useful. And there is no silver bullet to skip this process, you just have to do it. It's just a matter of how efficiently (and cheaply) you can do it here, and going to meets, trying stuff in stores, buying used and selling to recoup costs are all valid ways to do that.
Then again to people not in the hobby, you seem like an obsessed maniac throwing money at frivolous things you don't need. And of course, they're right.
But time you enjoyed wasting is not wasted time.
