SysLog UDP Load Balancing

You're setup is essentially one-armed LB. You must have a SNAT enabled so that traffic returns through BIG-IP; the SYSLOG messages should be in the UDP payload anyway, so I don't think there would be a problem using SNAT.

Regarding your monitors, yes SYSLOG can also work over TCP 514 but is your server listening on TCP as well as UDP? Either way, I would suggest it makes more sense to have a UDP monitor in this case. Not that I like UDP monitors, but you might need to use it along with a secondary monitor (it's only UDP!). If you can ping or telnet to your servers from F5, then TCP and ICMP should work. Not sure why you want to use gateway_icmp as opposed to just ICMP.

A quick google, brings up this article: https://support.f5.com/csp/article/K6143

To check the LB of your pool, look at the stats.