Best way--and hardest work--is to just keep pulling the bermuda grass out. Water well, then start pulling when the soil firms and water has soaked in a bit. There are a number of herbicides that target grasses, however few of them have labels for flower beds. In some cases, the grass herbicides will also have some activity against certain broadleaves, so if you go off-label and just try something, you might get a few of your flowers as well. You might look at Preen in a small area as it is recommended for grasses, but don't know what it may do to the plants you want.

I don't know that plant at all. but if the ID is correct, it's goji berry, a currently popular fad food/juice for anti-aging.

I have used Roundup around the base of many trees and it has never harmed the trees. I would never (again) use a weed barrier like plastic. I believe the rule-of-thumb for mulching around trees is no more than 6" deep around the tree.

I wouldn't recommend that as the minimum temperature for poultry is 165°. You'll get worms*. :-) The point is that plants are fairly heat tolerant--they regularly deal with surface air temperatures in excess of 100 degrees and do it pretty easily. Surface soil temperatures can ride hotter than that for short periods of time and the plant manages. Seeds are even more durable. Some require 160° for a full day before they're assured dead. And even then there can be surprises. Wet soil is a great heat sink, but has enough mass and absorbs enough heat that you have to apply an enormous amount to get a significant amount of the mass to rise high enough to kill root systems. That's the reason that solarization is done over fairly long periods of time. Plus as tap roots re-sprout into the heat zone, the new sprouts are constantly killed until the root is tapped. * We're six months off the proper season for that, but still.