Large starts can still produce- tomato, peppers, etc. I think you would still get some production out of direct sowed zucchini and cukes. You can get beans, carrots, lettuce, radish, kale, chard, other greens, marigolds, borage, nasturtium, etc. And in mid-June to July you'll want to get in your fall crops like broccoli, cauli, parsnips, cabbage, kohlrabi, lettuce, kale, etc. and winter and overwintered crops like beets, kale, cabbage, broccoli, cauli, carrots, cold hardy lettuce, etc. So still plenty you can get in! :) If you can, start your fall/winter crops indoors for a head start.
As far as prepping the plot, I would rip out any flowering/seeding weeds and then just take a spade to it and flip everything over, cut it up a bit and then dump compost over the top and rake that together. The weeds in the strawberries, I would dig out with a weed digger or a small hand shovel (trowel). I would then mulch with organic matter (grass clippings, leaves, etc.), straw, or wood chips. Then plant into that!