Triumph T3 900 Cold Start/ Might Start

Hello all,
A not so enthusiastic first time Triumph owner here.
I've always wanted a Triumph, and went to look at a 95' Triumph Trophy 3 900 with 38,000 miles. The hour was early, it was 20 F, and I had driven 40 min with a trailer for the bike. So when I got there and the bike was already running I didn't turn around and leave like I should have, but rationalized the POs action in prewarming the bike.
After taking a test ride, and noting bad fork seals, etc. I bought the bike and did the paperwork and loaded the bike. At which point the PO told me if it is really cold out to use a squirt of starter fluid/ether.
After trailering the bike home and the day had warmed up to about 50F I tried starting the bike and could not. And of course the POS did not answer my call.
The next day at about 50F I gave it a shot of starter fluid, no choke, and a quarter throttle and she started. After several minutes of manning the throttle she would idle on her own. I then rode all over town in search of new spark plugs. Any time I got her up to speed for a mile or so the temperature gauge would bottom out, and if I stopped she would not idle on her own. Which I assume is a bad or missing thermostat.
I changed the plugs, started her up without fluid or choke, but a quarter throttle. Took her for a spin but noticed no difference, and she wouldn't idle after speed induced cooling.
The next day I couldn't start her. With about 1/8th choke she stumbles and tries to run, but doesn't. Using an old plug I pulled off the far right spark plug wire (which feels a little loose) put the old plug in and observed good spark, BUT I noticed that she was still stumbling and trying to start. After putting the wire back on the right plug there was no change to the stumble. I then tested the middle wire/coil for spark and got good spark, but there was NO stumble/ wanting to fire at all with the wire off the installed plug. Next I did the same thing with the left plug wire, and got good spark plus she was stumbling and wanting to start again with the installed plug not active. When running the engine is noisy...
So what is my next move? Do I go straight to valve adjustment? Pull the airbox and carbs for cleaning? Beat the snot out of the previous owner?
Sorry for such a long first post, and thank you for help in advance.

 

I will try some carb cleaner, and report back. The tank was full when I bought it which doubled the value... My plan is to drive the old gas out. Yesterday I got it started with a bunch of starter fluid and then took it out for a fun ride. I will definitely post pics, but it will be a few days. This morning I pulled a back muscle, and any trip off the couch requires crawling.
Bored to tears laying on the couch so I did hobble out for a couple bad pics.

 

Thanks for that XCaTel. I will definitely check the choke circuit.
As much as I like tinkering on old bikes and boats I have to admit carbs are my nemesis, but at least with this bike I have one carb less...
I'm afraid it will be a few weeks though. I am still laid up with a pulled back, and I am supposed to fly to Costa Rica in a couple of days for holiday. I will report back once I have pulled the carbs.

 

So I pulled the carbs, and found the air filter had the k&n air filter mod. It looked like this. Cleaned it. Gave the carbs a quick clean, and put everything back together as the sun was setting. Anxious to start the bike before full dark, and while someone with diarrhea of the mouth coated everything in range including me. I blew the sprag clutch.
Now the bike looks like this.