In situations where - consumer doesn't trust\control workstation and the network, can consumer safely do a potentially sensitive activities over the internet when https is used? Say make a payment or transfer confidential documents?

Example use case might be - when I use my employer's computer and am in a network controlled by my employer or I am in an airport and use publicly available desktop there.

Naturally IT administrators in these circumstances are in full control of CA management and could potentially use a proxy and sign the traffic between the proxy and me and in that case the browser will think it's all safe and show green.

My current understanding is that as long as I can verify that the cert isn't spoofed and is signed by a well known CA rather than my company or some unknown authority, I can be assured that there's a real TLS channel established between me and service provider and I can be sure that there's one eves-dropping on that traffic?

Now if my previous statement is correct what is the best way to verify that cert? I understand that I can just open it up and inspect the CA's name, but is there any extra steps I can take like export the cert I'm receiving from the https site and submit it to some service for verification?