Everything will be controlled locally over your LAN. The Hubitat Hub talks to the SmartBridge Pro via Telnet. Only the "Pro" model has Telnet functionality, which is why that specific model is required. Hubitat does NOT support Lutron Cloud-to-Cloud connectivity. Hubitat only support the local communcations option. It works great!
Yes, link all of your Lutron devices to the SmartBridge Pro and make sure they function correctly before even attempting to integrate them with Hubitat.
Follow the Lutron Integrator instructions from Hubitat. Take your time and have fun. Once it is set up, it works extremely well!
P.S. The one extra tip I would suggest, is to make sure the IP Address of your Lutron SmartBridge Pro does not change. I personally use my home Router's DHCP feature for reserving IP addresses for my Lutron and Hubitat hub. Keeping them both from changing their IP addresses will greatly reduce any chances of the two losing communications with each other. Also, it is advised to plug these devices into a small UPS, especially the Hubitat hub, to prevent possible data loss/corruption during power outages.