To address the security certificate message, that is semi normal. Computers generate a certificate to be used for RDP connections. It is used to encrypt the connections. It should have been issued to the PC when it joined the domain and as long as your machine trusts your CA (which it should if they are on the same domain), it is likely that the certificate expired. I get this message at work on a lot of machines and I just click on "connect anyway" as I trust the PC even if the certificate is expired.
That being said, you should be able to do analysis of this problem remotely. What are you looking at that you need to connect to the PC to work on? Checking the event log or starting services can (and generally should) be done remotely without the added overhead of starting an RDP session on the server. Look into the tool "mmc.exe" that comes with all windows computers. You can use MMC to verify the certificate as well. If it is expired, you just need to renew it.
As a thought, is the SQL instance internet facing? If it is presented to the internet, there is a chance someone got in and is changing these things.