I can see right off you are mixing/applying your paint too thickly.
I have not used the Liquitex brand before but try starting with this formula as it seems to work with most air dry paints I have tried. For every 3 drops of paint add 2-3 drops of Fluid Matte Medium (Liquitex or Golden brand or you can use the Emulsion from the RebornFX line which is what I now use) and then add to that 10-15 drops distilled water according to how diluted you want the pigment. If the paint beads up on the vinyl when you apply it, then you need to add more binder in (Fluid Matte Medium) to offset the amount of water you used. You can also add another drop of paint if you feel you got it too diluted. This is where patience and practice comes in hand with learning how your particular brand of paint mixes.
Use a wedge that is not porous. The Swispers sponges are more porous and therefore leave the little dots you are talking about you do not want. Paint the paint mix onto your wedge in a thin layer and then pounce it on the kit. That helps you not apply too much at one time.
As for the lint issue, that is a given when painting with air dry paints. Because acrylic paints and mediums contain binders (which act like glues) to increase their adhesion, they also attract dust. You just have to work with it and keep your work space as clean as possible. If you ever try your hand at painting silicone, you will find this factor is 10x worse, which is one of the reason I do not enjoy painting silicone.
As far as the paint not sticking well to spots on the vinyl, be sure the vinyl is washed very well in hot soapy water (Dawn) and rinsed and dried well. Once it is clean, wear gloves while handling/painting to prevent you from transferring oils from your hands to the vinyl. If the vinyl absorbs any oils from your fingers, it will not take paint.
The other thing is be sure to prime your vinyl first. I know some say you do not have to but this ensures the surface will take the paint well. You can buy Primer from RebornFX line or you can use Liquitex or Golden Fluid Matte Medium as long as you got the version that says on the bottle that it can be used as a translucent ground. Apply it with a wedge pounced out thinly. Paint the product onto your wedge in a thin layer and then pounce it on the kit so as to apply thinly.
Lastly, take a deep breath and relax while doing all this. I find if I plan Pandora piano instrumentals on my computer while I paint that it helps me to be in a relaxed state of mind so I can flow with my painting. Figure out what does it for you.
Hope this all helps you some!