If you using a F5 load balancer (sticky sessions) in front of the web-tier, can I have a shared two-tiered web application environment that can have web-applications with two different session management configurations? For example, web app # 1 uses In-Process session management and web app # 2 uses a separate session state server? Both web application sit in IIS in the web-tier and database sitting in the data tier.
Yes, the type of session management you use is determined/set per web app. There's no requirement that multiple apps on the same server (or AppPool) use the same scheme.
With sticky sessions, each client will always be connected to the same server / web app instance, so you could even have web app #1 on server #1 use InProc, and web app #1 on server #2 use StateServer. Not recommended, of course (nor are sticky sessions), but possible.