No, it's not really possible to completely prevent weed growth. Weeds are incredibly resilient—all they usually need is a tiny bit of dirt to sprout, so their eventual appearance is pretty much inevitable. However, you can definitely keep them at bay by getting rid of weeds as soon as you notice them. The longer you let them sit and grow roots, the more they'll spread.

Weeds can grow through rubber mulch, but it depends on if you removed all the weeds in your garden beforehand. Weeds already in the soil can grow even when using rubber mulch. However, that doesn’t mean that your garden will be littered with rubber and weeds, but it does mean that the occasional weed might still pop up. Another way you can increase your weed suppression is by using landscape fabric as a second layer. 

There are many options to eradicate weeds right down to the roots. Vinegar works, but it takes a week or so to do its thing, and the same goes for many commercially available herbicides. Boiled water is a quick way to kill off roots, or you can pull up weeds at the root.

If you spray a DIY vinegar solution that contains high salt, it can disrupt the soil's structure and make it difficult for plants to grow. That's why it's vital to avoid spraying vinegar solution on areas where you are trying to grow plants and flowers. When using a mix of vinegar and soap, some say you can plant a few days after the vinegar dries.