When Mars is on the opposite side of the sun from earth, can NASA communicate with a probe on Mars? ?

It's actually pretty rare for the the Sun to *eclipse* Mars - but, it does happen on occasion.  But, with the immense output from the Sun, there's about a 2 to 3 week window where the sun is so close to the line of communication that they place probes on 'stand by' until the orbits carry the planets into a more favorable area for radio communication again. 

Only if the signal is relayed from a satellite somewhere.

Yes, but not with a straight shot if it's directly behind the Sun; the signal would need to be bounced, which, compared to getting things into space, or getting them to land on Mars, is hilariously simple. 

yes, but only at night, hun. 

Too many unknowns to guess that answer:

Who-what was sending

What was detecting

Activity of the sun's corona, magnetic field

Any magnetic field or other disturbances on Earth

Spacing of the sun, planets (Mars varies a lot)

Angles of the objects in line of transmission

This situation is already quite a situation for communications to Mars probes and landers and data back from them.  Look up what NASA and JPL have experienced.

I would give you the contact for my cousin's husband, but he has retired from JPL early this year.  He was a manager of the giant radio antennas system for space probe messages.  He worked on many of those probe programs for more than 40 years.