Gas models use a separate battery pack for running the receiver and servos. Electric models usually use the same pack (via "battery eliminator circuit"), but there's a low-voltage cutoff that cuts the motor power off while there's still enough power to run the electronics. There's typically a "soft-cutoff" feature that allows you to restart the motors if you need power to land - you get maybe a 10 second burst of low power or 5 seconds at medium power before the cutoff trips again. It can help to stretch out an approach if you're downwind and not going to make the field.

Either way there is no technical problem with making a power-off landing. Such "deadstick" landings are very standard in the RC world, and most fields will want to see trainee pilots make several of them on a "buddy box" system under the control of an instructor before they will approve you to fly alone. They're no big deal as long as you're being responsible with where you're flying.

Radio range is essentially "as far as you have a line-of-sight". Typical radios will have a range of 1.5-3 miles but you'll lose visual on the aircraft at a fraction of that distance. Some micro-sized parkflyer aircraft have less sensitive receivers, but since they're smaller you'll lose visual at shorter ranges too. Some radios have longer range, particularly amateur radio operators who can operate with greater transmitter power. These are popular with the FPV crowd, but to operate on these bands you must have your callsign and address on the plane.

The long range can actually be a bit of a problem because the signals really will go quite far, and can result in intermittent loss of control if someone else is operating on that frequency. AMA-licensed fields have a pinboard (if you don't have the clothespin with your frequency on it, you can't turn your radio on), and the fields are spaced out far enough to avoid this (typically a minimum of 5 miles). You're supposed to check before you just go fly somewhere, but some people don't. 2.4 GHz spread-spectrum radios are getting very popular nowadays because they're largely immune to these radio hits.