Almost all dogwoods have opposite foliage. Your shrub appears to be alternate. Perhaps a willow???

The twig looks just like the willows I've been pruning today, and the regrowth after being cut down is also typical of willows. Good news is, you can propagate it easily without knowing the variety name if it's a willow. All you have to do is cut some branches asap while they are still dormant, before the leaves come out, and stick 'em in the ground and keep the soil moist (just don't let it get very dry, doesn't have to be actively wet all the time) for the whole first season.