In the everyday speech, we do not differentiate between them. Although, there are special words for hand and foot: кисть and ступня.
Does it really automatically mean "my" foot without using мій чи моя ?
Yes, the construction "у + pronoun in genitive (мене, тебе, нього, неї etc.)" means posession. You can say "у мене болить нога" or "моя нога болить", but the second variant is rarely used.
How would you say a have a pain in my feet? У мене болить ноги? Which was the other of the two options
How do you say I have a pain in my leg? When you have to complete a sentence with either нога or ноги without a translation