Personally I really like this shrub if you have the space. The leaves are architectural and shapely, it has the advantage of being evergreen, it is tough and shade tolerant and it produces fabulously fragrant chains of yellow flowers like miniature daffodils in mid Winter. (At least that's when they flower in the UK.) So if it's not invasive in your area I would rake up the fallen leaves for the dog's sake and then give it a chance for at least one year to see if it wins you over.

Looks a lot like a vine called Virginia Creeper, used in landscapes. Makes a very nice vine on walls or trellises. The leaves turn a beautiful red in the fall, then goes dormant for the winter. It grows an extensive root system and is very difficult to get rid of once well established. However, if you use some full strength glyphosate, not the ready-to-use mix but the stuff that is 40% or more active ingredient, you can use a cotton glove or paintbrush or sponge and coat new sprouts when they come out of the ground. This allows you to miss getting your holly by accident, and it will take the vine out but may need a couple applications as regrowth appears. The Roundup won't hurt the holly as long as you don't get any on the holly leaves even though their roots are close together. Your other alternative is to continue doing what you are doing now--keep cutting it off everytime some growth appears out of the ground to prevent any photosynthesis and gradually starve the roots out.

Looks (and sounds) like camomile, Matricaria recutita, to me. This is an annual so could very well pop up unexpectedly from some disturbed seed source.

You hadn't indicated in your original post that this was a year old crop. Obviously that makes a difference given it is a biennial crop. However there are varieties of the flat/smooth leaf spinach that do have pointed leaves that are NOT bolting. The smooth leaf Arrowhead varieties like Catalina, Indian Head, Space, and New Zealand come to mind. Serun - once bolting is triggered in the plant by either age or heat you really can't stop it. Dave