(Very) generic list of common healthcheck transactions -
ST22, SM37, SM21, DBACOCKPIT(DB02,DB12), SM13, SM12, SP12, SM50, SM66, ST03, ST06, ST02. XI(SMQ1, SMQ2,SLDCHECK), Our BW checks are exactly the same as any other stack...we don't go into RSA1 or anything like that.
We automate a some of this stuff via BOBJ and the Solman so that you get a very nice, formatted .pdf file. Some clients we check SPAD and SP01 (not many though). Apart from the mentioned tcodes you need to monitor SXMB_IFB to check stuck messages in comm channel for XI and SXMB_ADM for Adapter engine status. SLDCHECK also can be done if your XI system is your SLD. For BI RSA1 to check on the source system connection status for BW jobs.
There might be some critical jobs in your BW system which is critical for Business reporting, a quick check from SM37 is also helpful. The jobs can be identified after discussion with your BW/BI and Business . Further if you have solman 7.01 SPS 22 and above it will be a lot easier with BI and PI technical monitoring through E2E monitoring.