Need way more context here, but if I understand you correctly:

1) If a game's solenoid circuitry is designed to have coils without diodes mounted across the lugs (eg. WPC), then correct, the wires can be soldered to either side. Otherwise, the power wire gets soldered to the BANDED side of the diode. Power wires are almost always the thicker gauge wire.

2) From an electrical standpoint, no, a coil can be mounted either direction, however, some mechs and brackets are designed for a coil to be placed in a particular way. It's also in the best interest to install a solenoid with the lugs on the opposite side of the coil stop to reduce vibration at the solder points--this is particularly important on flipper mechs.