which REST API is available for invoking using curl "force offline of node members" and checking the "current connections" for the node member.

I assume you want to perform 'tmsh show ltm pool <pool> members' equivalent. The iControl REST endpoint is /mgmt/tm/ltm/pool/<pool>/members/stats: e.g.,

curl -sku <user>:<pass> https://<host>/mgmt/tm/ltm/pool/<pool>/members/stats

Generally speaking, to run the "show" tmsh command equivalent, append 'stats' to the end of the base endpoint. For example, for /mgmt/tm/ltm/virtual (list ltm virtual), /mgmt/tm/ltm/virtual/stats (show ltm virtual).

You can find the references to iControl REST APIs in iControlREST Home. Toward the bottom, you will find PDF documents. Pick the one for your version. On the left hand side, you will find the details of each endpoints. For example, APIRef_tm_ltm shows the endpoints under /mgmt/tm/ltm. The document is not user-friendly, but once you understand a way to play with iControl REST, you should be able to find the information you are after.