That's a tough one to diagnose with the limited info you've supplied. What would help is knowing the make and model of the HTs as that will tell us how sensitive the receiver is supposed to be and the conditions under which the failure occurs. How far away from the repeater are you and what is around you. It will also help if you can tell us if you have a watt/swr meter and have checked the power and SWR from your HTs to their antennas.
There are multiple sources for connectors besides the big A and Flea-Bay. Look for ham stores for ready-made cable assemblies and adapters along with various component suppliers such as Mouser and Amp.
I have one, and the major complaints I have are the menus and the fact that it can't monitor two channels at the same time. No APRS is also a negative, but for the price I paid (free from a drawing), I won't throw it out. It sits in a Go-Box where it might get used a couple times a year when I'm at an event.
Yes, feedback from the speaker to the microphone. The cure is don't try to listen to your transmission while talking. That's the reason for the feedback.
As far as I have been able to understand from the regulations, they only state you need to identify with your call sign and the period you van operate until you need to ID. There is nothing in the regs stating when you ID, only that you must.
You can't boost the output of the radio to overcome losses downstream from the radio. The FCC limited the radio's output to 50 watts so you have to eat the losses.
Check Amazon for a Mean Well Supply. These are as good as you can get without the high prices charged for ham radio supplies. Many public safety agencies use Mean Well supplies in their repeater locations - and solid, clean power is often a critical life safety issue.
A lower-gain antenna works better in areas where the signals can reflect from many surfaces (multipath). Gain works two ways, not only by increasing the radiated output but also by increasing the received signal. In an urban environment lower gain helps reject multipath signals, while in a mountainous area, it reduces the multipath reflected from hard walls. YMMV
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While the regulations agree with you, human nature would give more weight to the person being directly targeted. Showing the interference happened every time or most of the time they keyed up would show the interference being targeted rather than generally being the south end of a north bound donkey which carries more impact.
It can be reported to the FCC as an intentional interference but the person being targeted would have a better chance of getting action than another user being blocked from the channel.
What mode did you select for the squelch tone? To set the radio so it transmits tone only you need the mode as TONE with no receive tone. If you set the mode to TSQL the radio automagically assigns the transmit tone to the receive.
Several in the ham clubs I belong to have the Cotre 001D and the 004D we use on our local DMR net. They aren't the best on the market with a very low power output (around 2W) but they can access the local repeaters most of the time. They don't replace a good DMR radio, but they are fun to play with.
The accepted usage ratio is 80/20 with the radio transmitting only 20% of the time. So for 48 hours of backup, you would expect to draw 15 amps for around 9 1/2 hours, which I feel is high for a GMRS station. I would set the transmit figure to around 8 hours or 4 hours per day for a total of 120 A with 80 A on receive.
Are you trying to use a repeater or contact someone directly?
Repeaters must be programmed in designated repeater locations on most GMRS radios. These are usually locations with an R designation on the channel number.
You are correct in that this repeater owner has no right to tell someone they can't access those frequencies. If the users are using the repeater for more than an incidental contact, then the owner does have the right to request they stop, or he can turn the repeater off for a period. The rules are a little murky regarding private repeaters, but the uncommon solution would be for the repeater owner to use some common sense regarding the fact the repeater is on-the-air to begin with.