ICC Printer Profiles

The short answer is that Elements will recognize common embedded profiles, not only aRGB or sRGB, but also documents with Prophoto coming from Lightroom or Photoshop as well as document with already embedded CMYK profiles. The embedded profile will be used as your working space and the color management will take it into account for the dispaly and your edits. If the profile is embedded, your Elements color settings (optimize for print or display) are ignored. 

Then, when you want to print, that embedded working space is used, and you have to hope that your printer can recognize this profile; to be sure, you'll prefer to voluntarily convert to aRGB or sRGB. Home printers and advanced printing services will  be able to understand the aRGB profile and convert it to their own printing profiles depending on paper and ink. Consumer grade printing services may not be able to understand aRGB, in which case you have to convert sRGB.

Your question is also if Elements can convert an image into a 'custom' printing profile. The answer is no. That's not important, your home printer driver as well as your printer service will manage this. Some printer services (Dry Creek for instance) may provide an external conversion tool for that, but I don't really think it's worth the trouble.

The real interest in having the ICC printer profile is if you can 'soft proof' your image on your precisely calibrated display (with wide enough gamut). This makes the display show as much as it can the printed colors; you'll see where some colors won't be exactly rendered and you'll be able to choose a 'rendering' intent to compensate to your taste.

Elements does not provide soft proofing, but an additional add-on like Elements+ has a script for that.

https://helpx.adobe.com/photoshop-elements/kb/color-management-settings-best-print.html

Edit:

Also keep in mind that raw files don't have color profiles. You can't assign them from the ACR module, they are automatically created from your choice in the color management in the editor: aRGB if optimized for pring, sRGB if for display.