When was it planted? It looks healthy to me although you planted it much too close to the structure. Crepe Myrtles need space and plenty of air circulation.

It was planted a week ago today. Again, I really don't know much about planting, but wanted to take on a project for the new home, and hoping to keep the tree alive.

It looks pretty good for only being planted a week! For this heat season, make sure that you water when it looks like it's about to be wilted and try to keep the water off of the foliage, in other words, water at the soil level.

Crepe Myrtles are susceptible to powdery mildew. They need lots of room for air circulation. Very hardy choice for a beginner. Is it next to a fence or a structure?

As PKponder noted, it looks as if it's too close to the house. Check to see how tall and wide this plant grows - not familiar with Catawba, but my crepe myrtles were tall and wide. Also, plants tend to stress in the mail - all wrapped up in a box - not a natural environment, so you can expect some leaf drop or wilting in the day or so after receiving it. Easy on the fertilizer. I would expect the dirt the plant came with (unless it was bare root) to be fertilized, so I don't usually fertilize a new planting for a few months. If it doesn't perk up, then investigate. Also, I don't usually plant the plants I receive in the mail right away. I keep them with their soil in pots in a shady /dappled sunlight area for a week or so.

The structure behind it is a fence. there is more grass on the opposite side of the fence. How often should I water it at this point? Also, should I apply new mulch around it?

Agree with PK, it looks fine. You don’t have to overthink or obsess over these, trust us! Just keep it watered the first 1-2 yrs, til established. The amount will depend on various factors — amount of sun, soil, wind, temps, etc. Stick your finger in the ground and see if it’s dry. I’ve grown them over 25 yrs in heavy clay, no amending of soil is necessary. My newest one is a few weeks old now, and I water it once a week if no rain. More mulch is fine, just not up against the trunk.

If your winters are extreme, that will probably keep the size in check. Otherwise, yes, Catawbas can get nearly as wide as tall. You might want to ask at a local nursery if CMs die back in your area in winter ... if yes, it should send shoots out the next spring.

If if you want it to be tree form, go ahead and snip off the lower branches. If not, leave them. Either way is fine, personal preference.