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As long as you have gotten the majority of weeds out of the area and put down a thick layer of mulch (about 2-4 inches), you should be able to control your weed problem. However, you could also place layers of moist newspaper down before adding the mulch to kill any weed seeds that may be present.

Mulch is a great way to keep weeds at a minimum in the garden. The following articles should be of some help to you: https://www.gardeningknowhow.com/special/organic/weed-it-and-reap-how-to-make-your-garden-more-appealing-without-chemicals.htm
https://www.gardeningknowhow.com/garden-how-to/info/whats-the-best-mulch-for-my-garden.htm

Here is an article that you may find helpful: https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

One of the best ways to keep the garden weed free is to place some type of weed barrier around plantings and within pathways such as plastic or fabric sheeting, or moist newspaper followed by a layer of mulch. Good options for mulch in the veggie garden include shredded bark or straw.

If you could provide a little more information (i.e. leaves, color, blooms, vining habit, etc.), we may be better able to help with identifying it.

There are either broadleaf or grass specific weed killers that will not kill vegetables. They are sold just as that - either broadleaf weed killer or grass weed killer. You should be able to find them most places that weed killers are sold.