Hi,
Yes, an 7812 will pass lower voltages, though with a bit of voltage loss. (I'm speaking from experience here; this thing has bitten me once ;-)
E.g., you have 10V going into the 7812; at the output you will have about 8V (+/- 1V). I think the same applies for lower voltages than 10V, but I have no proven data on that. You could of course give it a quick try yourself, by varying input voltage and measuring output voltage. Quick & easy, I'd say.
As to driving LEDs: until recently, your setup (voltage regulator & individual resistors for the LEDs) was how I drove LEDs. You can also rig a 7812 to be a current source: instead of a constant voltage, it supplies a constant current (say, 20mA for a LED). But before I'll explain, I suggest you search the web ('7812' 'current source' should do the trick), or that you give us more information about your setup & your needs.
BTW, if you have 12V, you could put several LEDs in series (about 2-3 if white LEDs, which need about 3.5V each; about 10 red LEDs, which need about 1.7V each) and use only one series resistor for each LED-chain. That way, you lose much less power in your series resistor (which can also be much smaller, perhaps only 20-50 ohm) and get much more light. This is more effective (& simpler too!) than using, say, 3 individual LEDs, each connected to 12V via its own resistor.
If what you really need/want is a voltage regulator for your battery (unclear to me), you should have a higher voltage than 12V; you could use an LM317, which you can easily adjust, or you could use your 7812 and put a few (2-3) diodes in the middle leg to ground (kathode to ground). This would raise your ouput voltage by .7V for each diode.
So, in short, I'd advise to put at least several LEDs in series for each chain; and the 7812 will protect your LEDs (& batteries, but not really a problem) against voltage spikes (as long as they are lower than 35V, which I think is the maximum voltage the 78xx-series is capable of handling.
Good luck & let us know if you need more specific info.
Peter,
The Netherlands.