has anyone here looked into using the proprietary (synopsys + cadence) tools or are we solely allowed to work with open source tools for this shuttle?

For SNPS tools, you would need sky130 ipdks. Not sure if it's opensource

I was under the impression that the OSU cells can be used with SNPS but not sure if they also used non-open source stuff in that

Analog - you need ipdk Digital - there are some hacks with which you can make ICC2 work I have done that using nangate 45nm, but its not accurate https://github.com/kunalg123/icc2_workshop_collaterals

ah ok