To answer the title question: no, a professional microphone cannot be plugged into a computer to produce in any way acceptable results. For condenser mics, this doesn't work at all because they need 48 V phantom power, and dynamic mics have too low signal levels for the low-quality on-chip mic preamps that are used in virtually all consumer-grade sound cards, so it does work (using an XLR -> 3.5 mm adapter, which is, like, never done) but with a horrible SNR.
However, this question isn't really relevant because what you have there is not such a microphone – you obviously do not plug in the mic itself but the receiver, which is for the purpose of recording just an ordinary line-level device and should as such offer easily enough signal level to feed any input.
It should, in principle. In practice however, my experience is that recording over consumer sound cards is always lottery, you must expect to end up with unacceptable amounts of static, strange distortions etc.. Often these problems can be solved with DI-boxes, but the only real solution is to use a proper audio interface. Most common these days are external USB interfaces.
How do I record using a microphone?
I'm trying to record some music onto my computer by trying something along the following lines - getting a microphone that I can connect into my computer and using it to record sounds from my un-electric acoustic guitar and some finger tapping/creative drumming. I know it might sound like a stretch but I have a good idea of the sound I want. I've been doing this for some time while just jamming, I just dont know the technical details concerning recording it like what kind of microphone (condenser or stereo I dont really know much about any of that stuff), or what kind of cables or if I need a mixer or what ever the heck an XLR is that my roommate was telling me about, or the software that I would need