I'm a bit confused when you use the word "box"? In some instances you seem to be referring to the circuit breaker panel and other times to a box where the connections to the utilizing device would be made.
Voltages of 220 and 110 have not been in common usage for at least fifty years. Today the voltages are 240 or 120.
The first line is the old water heater line (nased on the inscription in the box). It is 220. I was expecting to find three wires (2 hot and a neutral). I found two. Using a tik tester, both wires respond as hot. My questions:
1) is this likely stray voltage, and one of the wires is neutral? If so, how do I check?
2) shouldn't there be two hot wires if it is 220?
3) Is there something I can do to wire a 110 to this without opening up the box?
Electric water heaters use only two "hot" wires and an equipment ground wire. It is highly unlikely that either wire is a neutral. You would use a voltmeter or (better for amateurs) a solenoid-type voltage tester to determine the voltage. Assuming that when you write "box" you mean the circuit breaker panel the answer is no, it would require that you find the corresponding wires in the panel and reconnect them so that one of the wires is connected to a circuit breaker and the other wire is connected to the neutral bus bar.
The second line has three wires (two black and one white) Two of the wires are registering as hot, and for once, the white is neutral. The issue is that I have to shut two breakers down in order for the wires to stop setting off the tik tester. My questions:
1) once again, is this sray voltage? If so, how do I figure out which of the breakers is the correct one?
2) given that I have two hot, is this 220? I Haven't hooked up my multimeter to check yet, but thats my next step.
3) If I want to use this as 110, can I just use one of the Hots and leave the other capped inside the box, or do I need to trace it back to the box and disconnect it? I'm not excited about opening up the box .....
It MAY be "phantom" voltage or it may be the full voltage, it depends on if the circuit breaker was "on" when you tested. Again, the best test is a solenoid-type voltage tester. As for which circuit breaker is the correct one...they are both correct in that one CB feeds one "hot" wire and the other CB feeds the other "hot" wire. By rights, these circuit breakers should be side-by-side and in accordance with current code they should be a single two-pole circuit breaker with common trip and tied handles.
You CAN use this for two separate 120 volt circuits by using one hot lead along with the neutral for the first circuit and then using the second hot lead and the neutral for the second circuit. This is known as a "multi-wire branch circuit" and while not always desirable in residential wiring it IS code-compliant when done correctly.
ALL of the circuits when used for convenience receptacles need to be protected by circuit breakers rated at either 15 amperes or 20 amperes. The wire size must be no less than #12 for the 20 ampere circuits. Electric water heaters are commonly wired with #10 wire and have 30 ampere circuit breakers but years ago smaller water heaters were sometimes wired with #12 and had 20 ampere circuit breakers.
GFCI receptacles do not need an equipment ground to function properly but if they do not have the equipment ground then they must be labeled as having no equipment ground. Labels are included in the box the GFCI receptacle is sold. BX cable may, under some instances bu suitable for equipment ground but in most instances it is not suitable aas the equipment ground.