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Repeater owner uses odd codes/settings for the repeater and you pay a subscription fee for access.

Michael, receiver antennas don't need tuning as they are all band from one quarter inch stubs to mile long wires. Yes, the length matters in that a longer wire is more sensitive; but length doesn't affect the frequency range. You can prove this to yourself by using a simple piece of coat hanger or any wire connected to your radio and listening to the various stations on different bands.

If the radio is GMRS transmit only I wouldn't buy a dual band antenna but one for GMRS. You can receive on almost anything as an antenna.

To make a complicated subject a little clearer, the intelligence in an FM signal (GMRS is FM) is carried by the shifting of the frequency within its channel. Wide band signals can shift a total of 5 kHz while narrow band signals can shift only half that amount. The result is two-fold, the signal sounds softer and additional channels or paths for transmission can be created between the wide band channels.

The application requires certain personal information and the person submitting the application certifies they are the applicant so no; you can't buy a license for another person. That does not mean they couldn't make the application and then you use your credit card to pay for it. It's the statement where the applicant certifies all the statements are true. Falsification is a federal offense.

Could also be something as complicated as the volume knob....

440 MHz refers to the ham 70cm band which is below the GMRS frequencies. A J-Pole cut for the 440 band would need to be shortened by a small amount to put it into the middle of the GMRS band.

Why not ask your Ham club if they want to install it?

Truly the best solution is a rig built as a repeater rather than a home-brew setup. The key piece is the duty cycle of the repeater transmitter. It has to be active for both people and that means it is transmitting throughout the entire conversation. Your typical base station is designed around a 20% duty cycle which means you spend 80% of the time listening and less than 205 of the time talking. A 5 minute conversation is only a couple minutes for each radio but double that for the repeater.

You can use this to calculate the RF exposure. It's now required by the FCC. RF Exposure Calculator (arrl.org)

Narrowbanding was mandated for the VHF frequencies below 174 MHz.

PL-259 will mate with the female SO-239. This is the same connector pair used on mobile radios.

Welcome Tim. GMRS/FRS is a viable solution for your situation. A small mobile unit set up as a base station at your home with an attic antenna would guarantee communications between the two houses. A 20 to 25W mobile is all that's needed (actually 5 watts on an attic antenna would probably work). Midland makes a 15W unit that would be perfect. Look on Amazon for a small 12V DC power supply with at least 15A of output and you're golden for the base. A J-Pole antenna is what I'd use, there are several different ones available. Here's one made from copper pipe rather from twin-lead plastic cable: 462 MHz Land Mobile (GMRS) J-Pole Antenna - KB9VBR Antennas (jpole-antenna.com)

I believe it's called the ping-pong effect when one repeater starts keying another and they bounce back and forth. Other than different codes or powering down there isn't a way to stop it. Cell companies had the problem when private repeater nodes started appearing. Thet could shut the feed down to an antenna and then send the FCC to cite the repeater operator for an illegal transmitter.

Comet, Diamond, Nagoya are just a few that make replacement/upgrade antennas.