what does "one determined father" mean?

What does it mean by something burning low?

If something is "burning low," it means that it's running out. Typically with something that requires some kind of fuel, energy, or resource to keep going, and usually something with heat. Here, they're saying that the light is fading and going to go out. Ex: The gas for the stove is burning low. Ex: The candles are burning low. We should get some new ones.

Is definite article "the" used before "baby" to emphasize the baby?

The definite article "the" is put before "baby" because he is referring specifically to the noun of the baby. It doesn't necessarily emphasize the baby in this case because it must be said this way anyways for it to be grammatically correct. The article "the" always comes before nouns when they are specific or already known. If the noun is unknown or more general, you can use the indefinite article "a" or "an".

What does "out" mean here?

"Out" here means outside of a building or a room. She is saying that a calf is outside in the pasture. Ex: The sign says to keep out. Ex: I need you to stay out here until we finish decorating. Ex: Get out of my room!

What's "give birth" here mean? Is it a commonly used expression?

"Give birth" means to have a baby. It literally means to push a baby out of a mother's womb. It is a common expression. Ex: My dog gave birth to 6 little puppies. Ex: She gave birth to twins.

Is "at" necessary here because it seems to be okay by saying "eat away profits"?

In this case, "at" is a necessary part of the sentence as the full expression is "to eat away at (something)." If we were using this expression in reference to food (to eat something heartily), then the "at" would most likely not be required. Ex: Everyone, eat away to your heart's content! We have lots of food! Ex: Continuous bad weather has eaten away at the restaurant's profits.