Have not had the pleasure of any 12.1" screens, I have done a bunch from 14.1" to 17". And it is true that some of the screen bezels can be tricky, but for the most part the ones I have dealt with: Dell,Acer,Lenovo/IBM,Gateway,Toshiba,Sony,Packard Bell,NEC,HP/Compaq,Sager/Clevo/Alienware, Have 4-8 screws under the rubber bumpers, and then you carefully pull it off from the inside of the screen by the base,and work your way around,try not to snap too many of the clips,or ruin the double sided tape if present. For the most part, the inverter is in between the hinges, held in place by 1-2 screws, or none at all, has a plug on each side. Reverse it for putting it together, starting at the top, and making sure the latch system if it has one is working before putting screws in. (this last one is from experience.)
As for the screens, in the past 5 years or so, they have settled on a single plug style at the LCD end, this is to make them cheaper to produce, The same "style" LCD will work across the laptops using the same resolution and size screen. Glossy or matte makes no difference to the inverter or the laptop, they also are interchangeable.
Take for instance an hp that has 15.4" It could have a lp154w01(tl)(d1) or a ltn154at07, you could also find this same screen in a dell,Acer,Lenovo that is a 15.4" with 1280 x 800 resolution, it comes in either matte, or glossy. Try
if you are having a problem finding the correct screen. Also obviously LED and LCD don't use the same inverters. I have also dealt with Bliss computers for lcd screens, and have been very happy with them, they double pack and use about 10' of bubble wrap on the screen before boxing.
It typically takes me a hour or a little more to do a system board swap in a laptop, with everything else taking less time. And I agree it is a comfort level thing, I have been doing this for almost 25 years. And sometimes I forget that not everyone tears everything apart.

Always in trouble with the wife, because when she buys some new gadget the first thing I want to do is open it up and see whats inside. She still won't let me do the furby autopsy I want to do.