2. What happens with the regulator when the battery starts to lose voltage and it falls below 5V?
The linear regulator needs an input voltage up to 2v above the output voltage to output a stable 5v. The exact amount depends on the output current and temperature of the chip itself. The 7805 datasheets don't usually have detailed information about this as it's just a plain boring old common component with lousy specifications for today's chips, but other linear regulator chips will have graphs with the dropout voltage depending on the current output and possibly even temperature.
With the currents a 9v battery can produce, let's say up to 100mA, most 7805 regulators will have an internal dropout voltage of about 1.5v, maybe a bit less. That means when the battery's voltage drops below about 6.5v (but don't forget the voltage drop on the reverse voltage protection diode as well, which would be about 0.5-0.8v for a regular 1n400x diode), the 7805 will no longer be able to output stable 5v. It will output less than 5v and will continue to basically output input voltage - dropout voltage up to a threshold, let's say maybe 2-3v at the output.
For a slightly better linear regulator with decent amount of output current (800mA) and still very cheap, see LM1117 or a 1117 made by other manufacturer, here's an
example search at Digikey.
It's a linear regulator similar to 7805 but has a dropout voltage of less than 1.2v at low currents and temperatures as you can see on the datasheet (see page 7. figure
:
http://www.ti.com/lit/ds/symlink/lm1117-n.pdf .. It also doesn't need protection diode unless you go crazy with capacitance on output (like 1000uF or more) and the only thing you have to be careful about is that the pins on the regulator may be in another order compared to 7805 and that it needs 10uF or more capacitance on the output with some esr (no ceramic capacitor on output).
If you're sure your product won't use a lot of current, you can go even further and pick a linear regulator optimized for low current and low dropout voltage, like LP2950 for example which has a maximum output current of 100mA:
http://www.digikey.com/product-detail/en/LP2950-50LPRE3/296-20933-1-ND/1216905 As you can see on the first page in the datasheet (
http://www.ti.com/lit/ds/symlink/lp2951.pdf ) at about 60-80C (the regulator will heat up inevitably from regulating voltage because it's a small chip) and up to 100mA, the regulator will have a dropout voltage of up to 0.45v , which means excluding the reverse voltage protection diode, your device will get 5v even when your battery goes as low as 5.5v