I think your first tree may be Ulmus glabra 'Lutescens', golden elm. Here is a link that might be useful: Ulmus glabra 'Lutescens', golden elm

A close up of the leaves would be very helpful. It's hard to see what they look like in this photo.

Here are more photos, sorry, I know that they are large but they might make it easier to identify if anyone might know what it is. Here is a link that might be useful: More Mystery Tree Photos

'Should I be pruning it?' You only need to prune ornamentals if there is a good reason, eg dead wood, disease, poor shape, etc. Since your tree is healthy and beautiful and has remained so for 7 years without attention there is no reason to start chopping at it now.