Improving my new speakers sound quality

Add a piece directly behind the speaker. Then raise up the plastic vapor barrier and apply a piece to the upper portion on the door. It will be a large opening that you can fit some in. Also add a piece to the lower portion of the door.
You don’t have to wait for it to be warmer, I used a heat gun to warm the sticky side of the material to apply to the door. If you don’t have a heat gun, use a hair dryer. You also want to clean the area you are applying it to first. You can use a degreaser or just a cloth with soap and water. I used a cheap bottle of Awesome to spray on mine to clean it.
You can check my previous pics to see the locations I added it to. You def don’t need complete coverage

 

I will preface my response by saying I am neither an audio nor speaker design engineer, but I have built more than a couple speaker enclosures, both sealed and ported, for home audio as well as a couple for cars. Radio Shack put out a couple of good books (and I read them and still have my copies) on designing and building speaker enclosures back in the day and that information still applies now. The answers to your questions...

Short version -1) more is better, you don't have to do everything on the inside of the door but you do want to do more than just behind the speaker. 2) Doing the surfaces of the inner door certainly can't hurt, but covering the holes you will have to decide for yourself and 3) 60 degrees should be plenty warm enough.

The long answer... (and I'm open to the possibility that I might be way off base on some of my thinking - it has been about 10 years since I build my last vehicle speaker enclosure, my father-in-law just had the bare speakers sticking to the walls of his panel van by their magnets so I designed and built an enclosure for them that made a huge huge difference).

As to coverage in the inner door, it depends on your goal. If you are going for sound deadening only, then supposedly 40% or 50% coverage on the inside of the outer door skin is enough to notice a difference. I did closer to 60-70% coverage and while I did notice a difference, it wasn't as much as I had hoped. The biggest difference I noticed was simply the installation of new speakers (Infinity Reference 6x9 front, Infinity Reference 6.5" rear, $201 incl tax from Crutchfield with all the wire harnesses and mounting brackets). The idea here is that you add mass to the door panel which slows the vibration of the panel when the sound from the rear of the speaker bounces off it. This lowers the resonant frequency of the door panel and helps tame the muddiness of the sound. So, I think the more the better, but there is a point of diminishing returns and that may be where the 40% or 50% figure comes from. It probably also depends on what you are using for sound deadening material. I am assuming you are using Dynamat, Noico, or some other peel and stick rubber sheet with a metallic skin. If you are using mass loaded vinyl, I think this also applies, but I don't have any experience with it and it has been a while since I have done any research on it, so I can't say for sure.

If you goal also includes insulation, then yes, do as much as you possibly can in there.

As for those big holes covered in plastic, I have not tried it, and in fact I recently watched an installation video where a guy did that. I've seen people cover all the surfaces of the inboard shell around those big holes and that seems like a reasonable thing to do, but it was the first time I'd seen someone cover those holes up with the stuff. I'm probably going to try that myself when the weather gets nicer here since I have plenty Noico 80mil left over from what I did earlier in the fall. I have a couple of thoughts on that, though. First, speakers with small magnets and large cones are really best used in a sealed enclosure. The reason stock speakers sound boomy is because the magnets aren't powerful enough by themselves to precisely dampen the movement of the cone. They rely on the positive or negative pressure difference between the inside and outside of the enclosure to assist the magnet in controlling the movement of the cone. So, for the stock speakers, sealing those big holes with rigid material would help remove some boominess, but also decrease the volume a little. However, with no rigid material (it is pretty flimsy once you remove the backing) to back up the sound deadening material, you are introducing a vibrating surface where there wasn't one before. It could introduce weird vibrations or shoutiness at certain frequencies (bad thing), or it may act as a passive radiator (not necessarily a bad thing), or it may just hang there because there are still air gaps in the door and the stock speaker isn't moving enough air volume to really affect it. But I think the biggest benefit to covering those holes is probably that it prevent sound from bouncing off the inside of the outboard (or outer) panel and coming muffled through the inner door panel.

Also keep in mind that that plastic is there to prevent moisture from getting to the back of the inner panels and causing nasty things to grow in there. So if you cover those holes, make sure you seal it at least as well as the plastic did.

60 degrees is plenty warm enough for installation. The Noico specifies an installation temp of 49F or warmer IIRC. Not sure what the other brands specify, but it should be similar.