Learn Hardcore Russian: Каждое утро я просыпаюсь рано.

Утро here is in the accusative neuter singular. Time expressions without prepositions typically take the accusative; neuter accusative forms coincide with the nominative, so утро appears unchanged.

How does каждый agree with different nouns, for example how would you say each day or every week in Russian?

Каждый is an adjective and must match the gender, number, and case of the noun it modifies. Common time-expression forms in the accusative are:

Каждый день (masculine)
Каждую неделю (feminine)
Каждый месяц (masculine)
Каждые выходные (plural)

Why is there -сь at the end of просыпаюсь, and what is the infinitive form?

The -сь is the reflexive suffix. It turns the verb into просыпаться, meaning “to wake oneself up” (intransitive). Without -сь, просыпать has a different meaning (“to oversleep” or “to spill”).

What is the aspect of просыпаюсь, and how would you express I will wake up early tomorrow in Russian?

Просыпаюсь is imperfective, used for habitual or ongoing actions. Its perfective counterpart is проснуться. To talk about waking up once in the future, you say:
Я проснусь рано завтра.

Why is рано placed after the verb, and can I move it?

Russian word order is flexible. Рано is an adverb modifying просыпаюсь. You can also say:

Я рано просыпаюсь.
Каждое утро я рано просыпаюсь.
to adjust emphasis.

Why is the pronoun я used here, and can I omit it?

Russian is a pro-drop language: verb endings already show person and number. You can omit я and simply say (Каждое утро) просыпаюсь рано, but including я adds clarity or emphasis.

Why does Russian use the simple present просыпаюсь for a habitual action?

Russian does not distinguish between simple and continuous present tenses. One present-tense form covers both ongoing and habitual actions; context and time words like каждое утро signal that this is a routine.