Whether to use LAG, or not, really has no bearing on your question as LAG just creates one logical link, of any kind we desire, between the 3850 stack and the 3750.
If you want to share VLANs/subnets between the two logical L3 switches, you'll want a trunk link between them (which can also use LAG).
If both L3 switches have all the same VLANs/subnets, and SVIs for all the VLANs/subnets, just having routing enabled will allow all to work (although there are other design considerations dealing with whether you want SVIs on both logical L3 switches, placement of STP roots, avoiding unicast flooding, etc.)
If one or both L3 switches have VLAN/subnets not "mirrored" on the other, or don't have SVIs for all subnets, then to route between those, you need static routes and/or an IGP. Assuming you have at least one VLAN that has SVI on both L3 switches, you can use it for a transit VLAN. (You can also use multiple shared VLANs for transit, but generally you only use one.)
In answer to your specific questions:
For communication I assuming that I can LAG two ports from stack switch and connect to LAG to Cisco 3750 switch and assign IP on each switch. Create the VLANs as written above. Create static routes via LAG IPs between switches. Is this sound doable as per reference from above solution
As noted, if you trunk between the switches, with what you've posted, you don't need static routes.
IPs and IDs for new VLANs should be diffrent or we can keep same as of stacked switches ?
Generally, IPs should always be different. IPs should be within the same address block if sharing the same network. VLAN IDs, if sharing and trunking should also be the same. However, it's possible to use different VLAN IDs, for the same network, if you don't trunk.
