You cannot read a passive 125 KHz Tag with a NFC reader, because as you have already guessed, they operate on different frequencies.
There are passive RFID cards operating on 13.56 MHz, however, and these will read just fine on a NFC-enabled phone.
If you read the datasheet, it gives information on the typical and maximum current consumption for the IC. It also gives power down consumption and other details. I'd also check the other documentation (app notes, etc) to see if there is more information/advice on power characteristics.
On page 3 note 7, it says typical current consumption is below 100mA. 60mA is given as the typical consumption for the transmitter supply, digital and analog ~7mA each, and the pin supply up to 40mA. With this info you can get a pretty good idea of what capacity battery you need for 10 hours operation.
Assuming maximum consumption, and less than 100% efficiency (e.g. regulator used) we can do some calculations. We'll assume 80% efficiency and 100mA continuous draw:
100mA * 10h * (1/0.8) = 1250mAh
This is probably very, very conservative, but gives you a figure which will certainly be plenty. However the only way to get very accurate figures is to run some tests yourself, since consumption will depend on how many IO pins you are using, other circuit activity, regulator efficiency, temperature, etc, etc.
I would probably look at using a 3.7V Li-Ion cell, since it's a good voltage, they are conveniently sized and there are plenty of options around 1000mAh (eBay, Sparkfun, Digikey, etc) There are also cheap charging ICs available (Microchip do some good ones) Of course it's up to you, anything that provides the required capacity would work, but you have to consider what is most suitable for your project (size, weight, cost, etc)
Best Answer
The answer is yes. The reader (phone) has to scan continuously to see if it is there. Even if this does not happen at the hardware level, it would be possible to continuously "ping" the reader coil via firmware, so again the answer is "yes" - this will continuously drain the battery.