Welcome! 
(I’ve moved this to projects so you can get individualized responses based on your own needs and preferences.)
We started out with SmartThings and after 15 months ended up moving our critical need use cases to HomeKit because of the reliability issues with SmartThings. But we still like SmartThings for convenience notifications, which it does better than pretty much anything else in its price range, and we keep hoping the stability will improve. So we have a lot of experience with having both HomeKit and SmartThings in the same house.
The short answer is that as of this writing, June 2017, Z wave and zigbee devices Will not work with HomeKit (although the zwave alliance says there will be a HomeKit/Z wave hub available in 2018, but no details).
Most companies make either a homekit version or zwave Or zigbee version of their device, so it’s one or the other. For example, Fibaro makes sensors which look identical, but one model can work with HomeKit and the other model can work with SmartThings via zwave and they aren’t interchangeable. The same is true for Leviton switches. The cases look identical, but one version works with SmartThings, and another version works with HomeKit. Schlage locks work with either HomeKit or SmartThings, but not both.
specific devices that can work with both
There are a few companies which make devices which work with both and where are you can use either control method throughout the day. As of June 2017, these included:
Lutron Caseta Devices and Serena shades when connected via the 2nd generation Lutron SmartBridge
Philips Hue bulbs and lightstrips when connected via the second generation Phillips hue bridge
Some IHome pocket sockets
Logitech Pop buttons when connected via the second generation pop bridge
Ecobee Thermostat (HomeKit-compatible version)
You can use a Hue motion sensor connected to a hue bridge to turn on A hue bulb, and then have either HomeKit or SmartThings or both notice that that bulb came on and then do you other things, but it’s kind of a clunky set up. Other than that, all the sensors currently available work with one system or the other but not both.
WeMo has announced that they are going to release a bridge device this fall that will allow existing WeMo devices to work with HomeKit. It’s likely that this would be the same as Lutron and Phillips hue, where the devices could work with both SmartThings and HomeKit at the same time, but we won’t know for sure until we get more details on the new device.
Any device which is HomeKit compatible and also has an IFTTT channel could get indirect integration with SmartThings that way, but there aren’t many of these.
Conclusion
For the specific devices you mentioned, the Lutron Caseta switches, everything works great as long as you have the version of the Lutron smartbridge which is HomeKit compatible. Both SmartThings and HomeKit can control the Lutron switches, both get status updates, switching back-and-forth is seamless. We use this at our house.
For other devices, you just have to look case by case. But there was a new announcement at WWDC this month that Apple is going to make a software integration option available going forward, so that HomeKit compatible devices will no longer require a special hardware chip.
For some complicated technical reasons, that’s not likely to mean direct smartthings/homekit integration immediately*, but it should mean that we will start seeing more and more devices come on the market that could be used with both.
.*. since you mentioned homebridge, I’m going to assume you have a technical background. So the reason why the software option won’t be immediately available with SmartThings is that HomeKit is still going to require that everything be on the same LAN as a security factor. And SmartThings is still largely a cloud-based system. Same reason why when you use homebridge you have to set up a “man in the middle” server. But we’ll just have to see what actually happens as things move forward.
. I’ve been hunting around for information and with everything changing month-after-month it’s really helpful to find up to the moment information.