1 Battle.net Account, 2 Player, 2 different Blizzard Games

What you are stating here would actually be against the Terms of Use, specifically the following section:

Eligibility.

You represent that you are an adult in your country of residence. You agree to these Terms of Use on behalf of yourself and, at your discretion, for one (1) minor child for whom you are a parent or guardian and whom you have authorized to play the Game using the license granted to you.

However, to answer your actual question, I do not believe that you would be able to log into both at the same time, because Blizzard limits the number of logins per Battle.net accounts to one (1). Additionally, it would be problematic, especially due to the recent addition of Real ID; which game would Battle.net report that your account is logged into (WoW or SC2)?

I will admit, though, that this is purely conjecture, and I cannot say for certain right now, because I have not attempted to log into both at the same time.

The above answer is partially incorrect. You can multibox wow accounts under the same battle.net account, using the same or multiple pcs. Battle.net does not limit 1 login at a time. However, I can not tell you if they actively cross reference IPs to detect logins from different locations (unless, of course, you report your account compromised).

Realize that it is a violation of ToS to share your license with others and could result in the termination of the battle.net account.