Learn Russian: Даже утром я работаю.

In Russian, times of day often change their ending to mean "in the...". The noun утро (morning) becomes утром (in the morning) all by itself. You do not need an extra preposition to translate "in" here.

How do I know to use работаю instead of the dictionary form работать?

Работать means "to work". Because you are saying "I work," you must change the verb to match the pronoun я (I). For regular verbs ending in -ать, you drop the -ть and add -ю.

Is it grammatically correct to change the word order to Я работаю даже утром?

Yes, absolutely! Russian word order is very flexible. Starting the sentence with Даже утром simply puts the main focus on the time of day, highlighting that it is surprising or unusual that you are working then.

How exactly does the word даже (even) work? Does it always start the sentence?

Даже usually goes right before the exact word or phrase you want to emphasize. Here it emphasizes утром. If you wanted to say "Even I work in the morning" (implying others don't, but you do), you would put it right before я: Утром даже я работаю.