Do you often omit the subject in the sentence? 

Sometimes I sometimes see sentences without the subject. Is it very common in casual writing? Often, I often am confused when I encountering such sentences. I don't understand it well grammatically.

Examples: 

Went to school and studied yesterday.
Ate many apples last night.

Is this only for past tense?

You also encounter this form in poetry as well. Using your example,

Went to school and studied yesterday.
Ate too many apples last night.
A ruse, perhaps to skip school the next day?

Going to school and studying all day yesterday,

Eating too many apples from last night,

Now, a perfect

But agonizing way to stay home today.