Two separate questions here. So let me get to the one which may be more important to you:
If you are using the official SmartThings/Lutron integration, You will not be able to use the pico remote to trigger anything in SmartThings. The Picos will be invisible to SmartThings and SmartThings will not be told about any of their button presses . 
They will still work with whichever Lutron SmartBridge you get, which means you can still use them to control devices which are controlled by the SmartBridge itself, mainly the Caseta wall switches and pocket sockets.
As far as the official integration goes, I know you can use the regular Lutron SmartBridge
https://www.amazon.com/LUTRON-L-BDG2-WH-Caseta-Wireless-HomeKit-enabled/dp/B00XPW67ZM/
There’s a good bundle price that includes the smartbridge, one in wall dimmer switch, one wall plate, and one pico:
https://www.amazon.com/LUTRON-P-BDG-PKG1W-Caseta-Wireless-Homekit-enabled/dp/B00MXCRAX8/
And I believe you can also use the SmartBridge Pro.
You can’t use any of the other Lutron control devices with SmartThings, they are for different systems.
There is also an unofficial community – created SmartThings/Lutron integration. which does make the pico button presses visible to SmartThings so they can be used more like a minimote. However, you will have to set up an additional separate device like a raspberry pi to act as a server, and to get access to the pico information you would have to have the SmartBridge pro model, not just a plain SmartBridge. And it’s technically much more complex to set up than the official integration.
So the short answer is that with the official integration you can use either the Lutron smartbridge or I think the Lutron smartbridge pro, but neither is going to give you any pico information in SmartThings.