Answering question 1: Antennas like any electrical circuit require at least 2 conductors, just like you can't power a train or a light bulb with only one conductor.
Radio transmitters and receivers use electromagnetic radiation (or in the case of the 455kHz Legacy track signal capacitive coupling) through the air to send and receive signals. The simplest ones use the literal ground as one conductor (reference) and electromagnetically radiated energy traveling through the air as the other conductor. The airborne signal picked up by a radio receiver is much weaker than the ground signal. Antennas are used to collect more energy from a weak signal traveling through the air. The ground circuit in the radio receives the earth ground part of the signal. If a receiver gets too much of one part of a transmitted signal compared to its counterpart, then it won't get a clear signal. The two halves need to be somewhat in balance.
The TMCC Legacy signal is similar. The Legacy Base transmits its signal partly through the outside track rails (via the U post connection) and the other half is sent from the Base through it's power supply ground plug into you house's ground wiring. When the half of the signal transmitted by the rails sufficiently overpowers the airborne ground signal, loss of clear reception by Legacy receivers occurs. The antennas in TMCC/Legacy switches and Locomotives are picking up the signal from the earth ground that then travels through the air. The wheels get the other half of the signal directly from the outside rails (this can be thought of as the reference).
The loss of signal issue is not uncommon with TMCC/Legacy layouts having a lot of track in a fairly large open space. To remedy this, some people run a separate wire from the house ground near their track to help balance the Legacy signal.
I you want to definitively confirm or refute this possibility as the cause of your switch issues, temporarily relocate the plugged in extension chord right over top of the switch that's being unresponsive to Legacy commands. If that switch then works, we can discuss next steps...
I'm not qualified to answer your 2nd question about LCS and ASC2s and their use with switches. Maybe @MartyE, @Railsounds or one of the other LCS gurus here on the forum could answer that question.