Growing after/ into plastic mulch?

When we moved to our new house, weeds quickly began to take over everywhere I looked. A massive septic installation had just been completed, with all the destruction, compaction, and junk fill that entails. We weeded diligently, but there is an endless supply of new weeds banked in the soil.

One large section is in progress as a vegetable garden, with a couple of raised beds, surrounded by perennial flowers... and formidable perennial weeds. Out of desperation, in spring 2017 I put down a 10×25' strip of tough black plastic sheeting in an unused area, to give me some breathing room. It's been there ever since, and is holding up well; there are only a few holes where things have poked through.

My question is this: how suitable is the soil beneath for growing, after this much time? It sounds like when people solarize, or simply plant into plastic sheeting, it's only covering the ground for a few months at most. Has the soil undergone drastic changes?

ASAP, I was thinking of cultivating and enriching the soil under the random holes (probably making a few more) and setting melon and winter squash seedlings there. They might appreciate the extra heat from vining over the plastic. Is the soil healthy enough to support them?

Next spring, I plan to add more raised beds where the plastic is. Should the soil in that area have time to "breathe" over the winter? Or simply remove plastic before planting and cover with leaf/grass mulch?  Can I trust that mugwort, pokeweed, ailanthus and other troublemakers are truly dead after nearly 3 years... Or play it safe and blanket with newspaper or cardboard?

I would love to hear any information or personal experience you may have on this subject. Thanks!

I have planted areas that I deliberately killed this way many times.  I've found that it rebounds quickly.  If I were in your position, I would

1) Remove all the plastic.  It may have held up well up to this point, but it is inevitable, it will break down and you will have a giant mess that is very hard, if not impossible, to clean up.
2) Make a hole where you want to plant squash, melons, whatever, fill it with compost, and plant in it.  I would plant the entire area, even if I didn't have enough compost to cover all of it.  
3) If possible, cover the entire area with wood chips.  If you don't have them, put down compost over as much of the area as you can and plant it immediately with anything you have to get a ground cover in.  The faster you get it planted, the better.

If you make compost tea, spraying the area will help get it back to living soil quickly.  Coffee grounds help a lot too.  They bring tons of worms as well as fertilizing.  Any organic matter you have can be used as mulch and helps tremendously.  I use grass clippings and any other green stuff I have, as long as it doesn't have weed seeds in it.  I don't put green stuff too thick because I don't want it to go anaerobic, but an inch or two will help a lot with keeping moisture in and keep the sun from baking the ground before your plants can take over.

Have you lifted a section of the plastic to check the condition of the soil underneath? After three years, the perennial weeds ought to have truely been smothered but I’d still put cardboard down and mulch before building your raised beds. If you add plenty of compost and fertiliser into the planting holes of your squash, they may do ok but would consider them a sacrificial crop to test the soil. A well meaning friend laid landscape fabric around my mother’s suburban garden and mulched over it with bark chips for about five years and none of the established shrubs did well, they all eventually died despite fertilising them regularly and when I finally persuaded her to have the landscape fabric removed, the soil was devoid of any life, not a worm nor insect to be found.