Yes the signal is compressed as it goes over the coax cable, but the reason that so many channels can be sent at the same time is due to modulation. You may already know this, in which case ignore the rest of this post.
if not.... A cable carries an electrical signal (obviously). The signal is composed of many different frequencies, just as sound contains different frequencies. The cable company allocates a certain portion of these frequencies for each channel. Since every channel only needs, say 4 Mhz (i have no idea if this is right, just an example) to be transmitted, many channels will fit within the frequencies that can be sent over the cable. (channel 1 in 1-4mhz, channel 2 in 5-9Mhz and so on). The hardware that turns a certain channel sent over the cable into a picture on your screen must operate at a specific frequency. So you need a tuner (NTSC, ATSC) to demodulate the channel you choose from the signal that arrives from the cable company. Basically, the tuner tunes in to the frequceny you select, filters out all the other channels, and resends the selected channel to the picture hardware at the frequency it needs. (rather like concentrating on your conversation despite being in a crowd of people). That's the general idea.
Also, if the channel is sent digitally, several channels can be multiplexed on the same frequency that would usually hold only one analog channel. Basically, they send channel 1 for a while, then channel 2, then channel 3, and so on, and cylce back to channel 1. If they do this fast enough, then the decoding can take place without any real delay, since tv is broadcast at 25fps (?). With digital channels, the cable company can prefix the data with a tag that says which channel it is. With analog signals, the same technique would be possible in theory (i would think) but you would have no way of telling when channel 1's information stopped and channel 2's began. Keep in mind the cable company still has separate frequencies to on which to broadcast multiplexed signals.
As for using the same coax in your home instead of hdmi... as mjones73 said, you can. You just have to do the tuning with a tuner inside the tv. Incidentally, if you go to radio shack, you can pick up a modulator. These are used for old tv's with no composite + rca audio inputs. you plug the composite and rca audio in and the output is an old analog signal that you can put into the cable/antenna input of the tv. you can usually select channel 3/4 to broadcast it on. also, remember your vhs player has a switch to select channel 3/4 to display it's own output on. this is all the same thing.
i hope this answers your question, though perhaps more long-windedly than you would like.
p.s. if you're sick of expensive hdmi cables and the like. try www.mycablemart.com