To elaborate, take the sentence
Hago la cama.
This means ‘I make my bed’.
However, the sentence
Hago mi cama.
Means the same. My question is are these interchangeable for all instances where the sentence goes [verb] [pronoun] [noun]?
This is a complex, yet interesting problem. Mainly, because among non-native speakers it's quite confusing.
To put it simple: Definite articles can be interchangeable with the possessive adjective, but it hugely depends on the context.
As discussed by Said-Mohand (2010) and King & Suñer (1999), in Spanish it's common to use the definite article as a possessive when the "owner" of the object can easily be inferred (contextually or using the reflexive). This is why we use the article instead of the possessive when talking about parts of the body. Look at these examples:
Me duele la cabeza translates as:
My head hurts.
¿Sabes qué tengo en la mano? translates as
Do you know what I have in my hand?
However, as discussed in the aforementioned references, we must use the possessive when 1) we want to talk about autonomous objects and ignore whatever relation it holds with the speaker, and 2) we don't have extra information on who is the owner of the object. Look at these examples:
El doctor hizo unas cuantas puntadas en su mano translates as:
The doctor gave him some stitches in his hand. In this context, the relationship between the
hand and the owner (
he) is not important, so we don't use the definite article.
Tu coche está allí translates as:
Your car is there. We can't use
El coche está allí and still translate as
Your car is there because
El coche está allí is ambiguous and we don't have any information about the relationship between the object and any of the persons taking part in this dialogue.
Going back at your example:
Hago la cama is not
Hago my cama. We would need extra info to properly assume that
la is used as a possessive.
Hago la cama.
is ambiguous, as it may mean "I make my bed" (in which case it is equivalent to: Hago mi cama) or may be used to mean that the person makes some other bed specified in the context.
However, unlike in English, it is usual in Spanish to use the definite article instead of the possessive to mean somebody makes their own bed.
Alternatively, a pronoun can be used to make clear you make your own bed:
Me hago la cama.
One can even hear the redundant:
(Yo) Me hago mi cama.
to mean that the speaker only makes his/her own bed, or makes his/her bed by him/herself. In this case it is usual to make the subject explicit.
Note: I'd like to clarify something after reading the other answer, mostly focused on parts of the body. When parts of the body are not involved, the use of the definite article is mainly due to the fact that the person only has one bed, or even one car: Tengo el auto afuera (meaning: My car's outside.) The singular feature does not apply to parts of the body.