Learn Turkish: Bütün gün çalışıyorum.

The "I" is built right into the verb! The personal subject suffix at the very end of the verb tells us who is doing the action. Here, the -um at the end of çalışıyorum means "I". Because of this suffix, you don't need to use the pronoun ben (I) at the beginning of the sentence, though you could add it if you wanted extra emphasis.

Why is the present continuous suffix -ıyor instead of -iyor or -uyor?

The first vowel of the continuous suffix changes to match the last vowel of the verb root using 4-way vowel harmony. The verb root here is çalış- (work), and its last vowel is ı. Following the rules of harmony, an a or ı in the root means the suffix will use ı, giving us -ıyor-. After adding the fixed suffix -um for "I", we get çalışıyorum. Note that the o in -yor never changes!

If I want to say "I work every day", can I still use bütün gün?

No, bütün gün specifically means "all day" or "the whole day" because it focuses on a continuous block of time during a single day. If you want to say "every day" to talk about a repeated daily habit, you need to use the phrase her gün.