Electrical – One light switch powers two separate light switches

Sounds like you want 3 and 4 way switches everywhere. That’s fine. Designing where you put switches always should come back to how people use a room and putting a switch where they expect it. When using a 3 or 4 gang box the switch closest to the door should be the main light, the one you use most often. Then the lesser switches are farther from the doorway. It’s all common sense stuff. The only trick is to have the imagination to know how you’ll use a room when all that’s around you are bare studs.

Bringing up motion sensors makes for an interesting twist. I am putting them in bathrooms more often now. My favorite use is to attach the undercabinet lights in a kitchen to the motion sensor. When someone walks in the undercabinets feel welcoming without overpowering. When walking in at night I don’t always want the overhead lights to come on at the brightness I cooked dinner at, but I’m always happy to have the undercabinets. If I went into the kitchen to cook I’m already trained to hit the switch for the overhead lights. So one option would be to put 3-way switches everywhere you’ll want them then next to that switch put a motion sensor that controls coved lighting in each room (its hard to say without seeing your house, but mine is from 1917 and every room except the kitchen has hanging picture molding and sometimes coved ceilings). I’ve seen that done well.

DON'T try random stuff when you get stuck

Trying to replace actual knowledge with "throw things at the wall and see what sticks" is a fatal error when dealing with electrical equipment. Why? The entire strategy is based on stopping when you find "the" combination that works. Actually, many combinations will work and also kill you. The only way to avoid those is skill, so the right thing to do is pause, research and measure.

I'll grant you if you are only connecting to 3-way switches, you can't go too far wrong - but nonetheless, this habit is so dangerous in almost any other context, that I want to address it.

Position of wires on multiway switches is useless

I assume you're changing switches and receptacles for color or style. In that, the most common gotcha (other than broken-off tabs on receptacles) is assuming 3-way switch builders do you any favors by keeping screw positions consistent. "They do not" is the understatement of the year. The only useful indicator is screw color.

Related, electrical wiring is not very well color-coded, and multi-way switch circuits are worse than not color-coded at all. I for one mark wires obsessively, so the next person has a chance of understanding the circuit.

Probably easiest to just "do it the old fashioned way". Travelers are always in the same cable (and are never ground wires obviously). Switch 2 is no help at all. So we look at switch 1. Only two are in the same cable (I can't see, you can) -- gotcha! Those are the travelers. Now, knock wood, that cable is continuous to switch 2, in which the same 2 colors will be the travelers there. (If it stops anywhere intermediate, all bets are off, and this is why I mark wires). Buy a 5-pack of colored tape, and mark all travelers yellow. There is no need to distinguish travelers from each other.

Then, travelers go on the brass screws. The remaining wire goes on black.

Edit: Looking at your photos in switch box 1, it seems clear the travelers are white and red. Wrap them with yellow electrical tape. Most likely they are also white and red at switch 2 (that's not 100% sure but it's the thing to try first.)

The light should have 1 terminal hot

Lights are the ultimate load, and in mains electrical, most loads connect between hot and neutral (unless you dealing with North American 240V or in the Philippines where everything is that). Therefore there should only be "hot" on 1 terminal, not both.

Edit: If the right lamp wire is energized, then most likely the 3-way set is delivering power correctly (it may still be wired wrong for the switches to work as intended).

If power is present at the right, and still it doesn't light, then it's a) burned out bulb. b) burned out socket. Or c) the neutral wire has a problem. There'd be no reason for the latter to happen, given that you don't touch the neutral when replacing 3-way switches.

Your next test is to fit the light bulb and test the left socket. If the left socket suddenly reads "hot", that means there's something wrong with the neutral wire. Otherwise you have a bad lamp or socket.

To answer your questions directly:

• If all six wires are hot, you need to file a warranty claim on your 3-way switches! The only other possibility here is that your voltage tester is picking up phantom voltage.
• Both sides of the lamp should never be hot. If it is, and the bulb is in it, that indicates a broken neutral wire. That left side is surely neutral.
• Neutrals don't have fuses. Their only overload protection is being monogamous to one hot, which is fuse/breaker protected. MWBCs are engineered to allow 2 hots to share 1 neutral. All other sharing creates overloads.
• Neutrals should never be hot. Travelers are in pairs; one will be hot. Switched-hot will only be hot when the switch is on. Always-hot is I'll give you 3 guesses. Notice I haven't mentioned a color yet. That's because colors don't necessarily correspond to functions. This is why I wrap tape around wires: in my world, switched-hot is red or blue, travelers are yellow in pairs. Here is how your box looks in my world.

(the thin lines are just to make clear which wire is which; the meat of what I do is the band of yellow.) The purple dashed line is a "Chinese Wall" between the left side and right side switches. Because nothing crosses, nothing is allowed to cross. That is relevant if you put a smart switch on the left: it must not steal neutral from the right.