This answer is somewhat outdated as Apple has added very robust and intelligent overcharging, over rate charging, temperature controls and adaptive charging to extend the life and avoid many of the situations where 2010 to 2015 era battery and charge system tech could result in less service length than ideal.
Even for 2010 era, what is described is still certainly safe - just turn it off and enjoy your trip. Newer gear makes this even more safe to do. You really have to go out of your way to exhaust batteries - and only random failures result in swaps in my experience now.
iPhone and all other Apple Products shipping in the last 5+ years (excepting the Apple Battery Charger which are NiMH AA cells) have lithium polymer (lipo) batteries. The MacBook Pro and Air batteries are a bit more sophisticated than the iPhone/iPod/iPad batteries but the underlying chemistry is the same.
It is safe to store them for months and it's best to charge them 50% and turn off the device for long term storage. All lipo batteries work best when they get a normal serive charge and discharge cycle once a month, but slowly draining from 50% to 0% over 6 to 12 months and then getting charged back to 50% is quite safe for several years of storage.
I wouldn't worry about leaving it off even 5 months if you starting over 50% but less than 70% charge in a not-hot location.
- The worst thing for a battery is to
use or charge it above 95 degrees F.
- The next worst is deep, deep
discharge (well past when the device
shuts off) - make sure you charge up
the battery past half before long periods of
- Repeated long storage while full.
(don't make a habit of leaving it
100% full and off for months at a time)
I still have my original iPhone going strong (for occasional testing) by waking it up every 6 months at a minimum. If I leave it around 2/3 full and wake it six months later - it's usually got more than 1/4 battery left. I then use it for a few days charging and discharging fully before shutting it off again.
Initially Apple used the phone number as the FaceTime "account" and non-user-visible SMS validation as the "password" to avoid forcing end users to enter/remember/forget yet another set of credentials.
The SMS validation happens behind the scenes - but SMS does have to be active on the phone at the time of first verifying (and potentially when re-verifying?) your FaceTime account on Apple's servers. If you switch off FaceTime in the Phone settings, wait a few seconds and turn it back on - you can see this checking process. The status message will change to say "Waiting for activation..." until Apple's servers verify your account. Once it says "Your phone number will be shared with people you call with FaceTime." the account is verified.
You can now use AppleID for credentials for devices that do not have SMS and an easily accessed/understood phone number. SMS is needed one time at activation to ensure Apple's push notification goes to the correct device - there shouldn't be a problem dropping SMS later once the device is registered.
This data lets the address book facilitate dialing and location over WiFi. Apple runs servers that let each device check in and request to initiate a call if the receiver is online anywhere. There is no technical reason why FaceTime cannot use cellular data for the call (bits are bits) but in practice the carriers would probably have to charge more for supporting everyone video chatting on networks that are currently straining to contain the data already flowing over them.
On devices that allow an email address - if someone calls you - all the devices currently on ring but only one picks up. For example, it's possible to receive incoming FaceTime calls ringing on a Mac, an iPod touch and an iPad - and any one can pick up the "call".
If your mom selected the 'notify when at this location' switch, your phone will notify her when it comes online at your house. It is highly unlikely that this is enabled, though.
However, this will show your phone as online, so if she happens to check Find my Friends while your phone is turned on, she will be able to see it on the map.