How a CMYK jpeg will display on a client's monitor is a wild variable. As Yorik notes, support for CMYK is spotty at best. Your system is much more sophisticated than your client's, and has color profiles installed that your client doesn't have. That's why the jpeg may look fine to you but awful to the client.
There is also a wide variation in how colors are displayed from one monitor to another. If the client isn't paying for an actual press proof, your best alternative is to make a color-accurate print and send that to the client with a note that total color accuracy is not guaranteed.
To answer your questions:
It will print close to what you see in Photoshop. Can't guarantee it will be exactly like that, because you don't have a calibrated monitor. If you have a good proofing printer and know how to use it with the "Photoshop manages color" setting, this can also give you a good approximation.
When you create a print document in CMYK in Photoshop and want to send your client a soft proof to look at on screen, you can do one of the following:
Save As a Photoshop PDF using the "High Quality Print" setting. Disable Photoshop editing capabilities, otherwise the file size will be huge. (Photoshop will save a complete copy of the PSD inside the PDF if you leave it enabled. That's what "Preserve Photoshop editing" means.) This is generally the best option.
Use Save for Web, which converts to sRGB by default, to save a jpeg. Send that to the client. Be sure to include the color profile.
Convert the PSD or your CMYK jpeg to sRGB using
Edit > Convert to Profile. Set the Intent to "Perceptual" and check "Use Black Point Compensation" and "Flatten Image to Preserve Appearance." If you convert the PSD, remember NOT to save the change!