Can the sun cast a shadow? The sun's light can cast the shadow of another object, but does it ever cast its own shadow?
 A: Even small regions of the sun can cast "shadows"; a sunspot appears dark because it is much dimmer than the surrounding photosphere, and also because it is "shadowing" the light from the hotter plasma deeper within the sun.
I once had the pleasure of observing a sunspot-like shadow here on Earth. I was at an art department's "iron pour" party, where they filled the molds for that term's cast-iron sculptures. Some molten iron landed too near the edge of the sand pit and started a little grass fire; the flames, viewed against the glowing iron, were dark.
A: Yes.  For example, on October 8th 1970 Earth was in the Sun's radiofrequency shadow with respect to quasar 3C 279.  In other words, quasar 3C 279 was occluded by the sun.  
Observation from just before and after the occulation permitted measurement of the bending of radiowaves as a test of general relativity. 
The sun would also block other frequencies of electromagnetic radiation including visible light.  
A: It could cast a shadow if there was a brighter light source behind it. Like a strong torch shining on a lit bulb.
A: We need to start by defining what is meant by "shadow." I believe that the more acceptable definition should be - a shadow is the result of the prevention of electromagnetic energy being sent by one object, from being received by another object. With this definition, anything that gets between the source and the detector, will cast a shadow.  In the case where the interfering object radiates the same spectrum and relative amplitude as the source, it will also cast a shadow, but it would not be detectable.  Therefore, the sun does cast a shadow when it interferes between a source and a detector.  
A: Yes.  A current example of how this is a concern in current science is in NASA's Kepler mission.  Kepler is a space telescope designed to look at a specific part of the sky and consistently measure the amount of light coming from the stars in its field of view.  When planets orbit in front of their host star they will create a shadow that decreases the amount of light that reaches Kepler; this decrease in light signals to scientists the very likely possibility of there being a planet orbiting that star.  However, planets are not the only thing that can create a decrease of light reaching Kepler.  Low luminosity stars that cross in front of a star that Kepler is monitoring are expected to be contaminants in the data.  From this website we read:
"While it has been estimated that 90% of the KOI (Kepler Object of Interest, or a star potentially having transiting planets) transit candidates are true planets, it is expected that some of the KOIs will be false positives, i.e., not actual transiting planets. The majority of these false positives are anticipated to be eclipsing binaries which, while spatially much more distant and thus dimmer than the foreground KOI, are too close to the KOI on the sky for the Kepler telescope to differentiate."
So, stars creating shadows by crossing in front of other stars is something that actually happens and is something that is a current concern in the planetary science today.  All you need is to have the sun cross in front of a larger and brighter star for a "shadow" to be made. 
One other thing to note:  the peak emission wavelength of the sun is in the optical range.  If the sun were to cross in front of an object with stronger emission outside this range (such as an even hotter star peaking in the UV) then the strength of the shadow that the sun produces will have a dependence on the wavelength of light used to observe.  For instance, if the sun crossed in front of a UV-strong star the sun's UV shadow from that star would be stronger than its optical wavelength shadow.
