I PLAN TO RUN AN EELECT. WIRE 100 FT. FROM MY HOUSE TO A STORAGE BUILDING. THE CIRCUIT I'M CONNECTING TO IS 12 GA. WOULD THERE BE ANY BENEFIT TO USING 10 GA WIRE FROM THE HOUSE TO THE SHOP?

Junction box is fine.
Must be accessible and not hidden inside wall.
http://waterheatertimer.org/Color-codewire.html
http://waterheatertimer.org/How-to-select-right-wire-nut.html

43 feet is not over 100 feet, so wire size can stay same.
Recommend 30 amp breaker and 10 gauge wire for 4500 watt water heater.
If circuit has 20 amp breaker and 12 gauge wire, recommend 3800 watt elements.
http://waterheatertimer.org/How-to-replace-water-heater-element.html

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Usually 50 - 75 ft. is as far as you want to go, especially with 16 gauge wire. What happens is the lights get dimmer as the length of the run increases. If you don't mind the lights getting noticeably dimmer, go ahead and add the other 4 lights. Using a heavier (12 gauge) wire will help a lot. You'll want the 12 ga. to start at the transformer, preferably running the whole length. Also, using a smaller watt bulb will help.

If you have a 120 volt outlet near the approximate middle of the new (total 150 ft.) run, you could relocate the transformer to that location and have two 75 ft. runs in 2 directions.

You could use 6/3 with ground. You must take an equipment ground conductor along with your current carrying conductors. Still drive a ground rod at the building, and add a seperate ground bar in the sub panel. Connect your equipment grounding conductor from main building along with the equipment grouding conductor to the ground rod. Do not bond the neutral in the sub panel to ground. Be aware you will still have some voltage drop, so a 220V air conditioner might be better.

The pump may be freezing up, but the wiring seems very inadequate if the power is disappearing. It should pop a breaker or fuse if properly wired, but overloaded. I would think that 12 ga wiring would be appropriate for less than 100 ft of wire with a one horse motor on 120 volt circuit. If you are on battery, then much larger wire and breakers are called for. Hope this helps!

Yes, just be sure connections are heavy enough wire, and obviously the inverter uses 12 vdc input. Hook jumper wires positive to positive, negative to negative, on all 12 volt batteries. Deep cycle batteries will work better than car batteries. Just remember, when re-charging them, that they last longer when charged slowly, so don't use a high amp setting on charger and to monitor them closely, and do it in a well ventilated area, away from any ignition sources.

I think he cost of the panel and breakers would be the cheapest thing on your list. Your 4100 watt genny is tiny for a whole house, and if you are planning on a solar set up, you will need to convert to dc, so, I take it your 4100 genny is used to charge the battery bank? Then the panel will not need to be anything but a small disconnect for the genny, basically one of those things that prevents the current from flowing back into the source, which you don't have. I would talk to the building dept., they may not want to take liability, and make you get a contractor to figure something out. All your appliances, and lights will be 12 volt, most motors are 12 volt anyway, they are just transformed to 12 volt from 110 current, and I don't think you will need breakers for 12 volt stuff. You could look at some camper supplies, boat stuff, they all run 12 volt appliances, and lights.

MMmmm, I'm doubting this has anything to do with wire. Can you test that theory by wiring one speaker with each type / length of wire and getting the same reslults?

Define the speakers and the amp for us.

Yes, It will work fine splitting it into 2 runs connecting them together in the transformer. It will work better than running them all on one line. Sometimes running them all on one line the last light s dimmer than the first light. I was an electrician for 14 yrs. and I promise you it will work.

no the problem is not the 10 gage wire , 10 gage is good for 30 amp circuits and you can always use it provided it is fused or breaker to accommodate the smallest wire in the circuit example , if a circuit contains 14,12,and 10 gage wire the fuse or breaker must be sized to accommodate the smallest wire which is the #14 which means it would be fused @15 amps but the circuit would work fine as long as you did not exceed the 15 amp rating on the circuit , you must have something wired improperly in the circuit re-check the wire connections