Hello MBJ, you sure have a problem there. The weeds you have are very pernicious and difficult to get rid of, especially the mares tails. I don't think you really have any alternative but to use weed killer on them and this will probably take several treatments.

As for using chemicals on ground where you want to grow, everything you buy in the shops has been treated with all sorts so you'll be no worse off. It will tell you on whatever product you buy when it is safe to plant and eat crops.

I would buy a weedkiller that kills right down to the roots - it will tell you on the labels. And a sprayer. They cost a few pounds but if you use a watering can you will get through much more weed killer. Mares tails are difficult to do as water just runs off them. The theory is bruise them first but you're inclined to get bits breaking off all over the place and I'm sure they all regrow. You could try wetting them with soapy water first and see if weedkiller sticks to them better. After you've sprayed and spray is dry cover them over with the tarpaulin. Don't expect everything to die instantly. It won't. You will probably have to repeat the weedkiller 2 or 3 times and keep covered for the best part of a year.

Sorry if this sounds pessimistic but it will work and can be done. It just takes time. Sure your just dying to grow something and see some results so pick a patch the size you think you can manage and just clear that bit by hand and start growing something for this year. It's not too late. Good luck and let's know how you get on.