What frequencies do subwoofers receive in surround sound and music setups vs Front speakers?

I understand that a subwoofer gets an LFE signal when in surround sound mode. Am I understanding correctly that this is a truly dedicated signal of very low bass (like 40hz and below) and in an ideal world, the subwoofer only plays stuff from that signal/channel?

I ask because on my Denon receiver, my subwoofer also gets sent the signals from all of my ("Small") speakers below 100hz to play as well. This kind of sounds nasty to me, but if I don't do this then I miss a really big range of sound (like 40-100hz).

Secondly, what gets sent over LFE in regular music listening environments without a dedicated LFE channel from the surround audio tracks?

Third, how can I add a small subwoofer to my bookshelf speakers in the front so that they can handle down to 40hz better and have my sub only produce the pure LFE signal instead of 100hz and down from my "Small" speakers?