Can I use the definite article for things which are somehow mine, which I can use my before them and they are unique to me (like my mother, father, wife, husband)?
For example to say
When I was 16 years old, the father said me .....
Instead of
When I was 16 years old, my father said me ....
or
As I fell, the shirt teared apart
To avoid ambiguity, you should use "my".
When I was 16 years old, the father said to me...
This is an odd construction. In some religious beliefs, it actually makes sense, though "father" would normally be capitalized:
When I was 16 years old, the Father said to me...
"the Father" can mean "God" and, in some Christian faiths, it is a term for a clergyman.
If you mean to say that it was your father, you must use "my".
You can also omit it entirely, though you would likely capitalize it here, too:
When I was 16 years old, Father said to me...
In this case, you're referring to him by his title (in this case, more of a name substitution), "father" and it's implied that you mean your own father.
As to the shirt example, it too can be ambiguous. If you want to imply that it's the shirt you are currently wearing which tore, it is more clear if you say "my shirt". The verb here should be "tore", the past tense of "tear".
As I fell, my shirt tore apart.
It is certainly possible to use "the shirt". For example, let's imagine you're in a fight with a sibling over a shirt. You each have one of the arms of the shirt and you're trying to get it away from your sibling. In this fight, you fall and as you fall, the shirt is pulled so strongly that it rips. In this case, you would certainly say:
As I fell, the shirt tore apart.
I would avoid using "the" for things that can be made more specific with "my".
When I was 16 years old, the father said me .....
This leaves ambiguity, "the father" sounds like maybe a priest or somebody else's father. It would make sense given the context of speaking about a father, but with no context it sounds unusual.
As I fell, the shirt teared apart
This, again, is ambiguous. In fact, it seems to slightly imply that the shirt which tore was not the shirt you were wearing. It sounds like a shirt you had in your hand ripped, while the shirt you were wearing was fine.
I would use "my" whenever it's appropriate to avoid uncertainty.
No. "My" is a determiner and takes the place of an article to make it more specific. When you use the definitive article "the" in this context it implies a kind of universality. So the phrase "the father", without a more specific modifier, is often used in religious context to refer to a creator deity, or as a title, or in some cases the founder of a cult. "The father of [noun]" refers to the creator or founder of that noun, so you might say "the father of the light bulb, Thomas Edison..."
Incidentally, in the phrase "my father said to me", "me" is an indirect object and requires a preposition.
Will makes an excellent point that "the" is technically allowable here, but is not preferred because it is insufficiently specific. I tend to let what I think should be done get in the way of what is technically correct. This difference is significant here because if you specified the subject in more detail earlier in the text, it would be allowable, and maybe even preferable.
So let's focus now on your second example
As I fell, the shirt tore apart.
Alone, this is not a good construction because it is entirely ambiguous as to what shirt tore. However, let's say in the previous paragraph you gave the history of your favorite shirt that your mother gave you when you were small, but someone took it from you and you are currently wrestling with them to get it back. Then, if you said
As I fell, my shirt tore apart.
That would actually be more ambiguous. Are you talking about the shirt you are wearing, or the shirt you are trying to get back? In that case, it would be better to use "the".
