Based on my experience, don't worry about the little elm that's dead if it's perhaps 3" across as it looks. The one that's not dead you want to either dig out or hit with brush killer because it will continue to try to resprout repeatedly. $800 seems awfully steep for such a small tree stump. You can get it out far cheaper with a shovel and a good pair of loppers and end up with a few tools for less than $100. Dig down, sever the roots with the lopper and remove. You can do the same with the dead one if you are worried.

I grow H. paniculata 'Quickfire', and it's not a small shrub. You will want to remove some grass and plant the Qf about where the edge of the grass is now, if I am judging size correctly. It gets to be 6 or 7' wide and while you could prune it to maintain size, not with hedge clippers. You would have to cut individual branches, a lot of work for a hedge, so I would plan to plant it at least 3 1/2' from the sidewalk so it doesn't grow out to block the sidewalk. Then you can plant a groundcover or annuals and mulch well for the first few years to keep the weeds to a minimum while the hedge grows in. (Mine was taller than I am at after 3 growing seasons from a rooted cutting.) You will want to do a weed patrol at least every couple of weeks since it's way easier to get rid of those baby elms or whatever when they are 10 days rather than 2 years old. It will only take 1/2 hour or less if you have mulch down.