Anoche tuve un sueño raro y me desperté temprano.

Anoche (last night) is a time expression, and Spanish often places time words at the beginning to set the scene. You can move it:

• Tuve un sueño raro anoche y me desperté temprano.
• Anoche tuve un sueño raro y temprano me desperté. (less natural) The most natural is usually either at the start or near the verb it relates to.

Tuve is the preterite of tener and presents the dream as a completed event (you had it, it happened). Tenía (imperfect) would sound more like background/ongoing description, often implying another event interrupted it:

• Anoche tenía un sueño raro cuando me desperté = I was having a weird dream when I woke up.

Both are possible, but they feel different:

• un sueño raro = a dream that was strange/weird (more neutral, most common)
• un raro sueño = a peculiar/unusual dream (more literary/emphatic) Adjective position can slightly change nuance.

Despertarse is commonly used to mean to wake up (yourself), so Spanish uses a reflexive pronoun: me, te, se, nos, se.
Despertar without reflexive often means to wake someone/something up:

• Me desperté temprano = I woke up early.
• Desperté a mi hermano = I woke my brother up.

Me desperté (preterite) treats waking up as a single completed action (the moment you woke). Me despertaba (imperfect) would describe an ongoing process or repeated habit:

• Me despertaba temprano todos los días = I used to wake up early every day.

Here temprano functions as an adverb meaning early and describes the verb me desperté. Spanish often uses the same form for adjective/adverb with temprano/tarde:

• Me desperté temprano = I woke up early. (adverb)
• Es temprano = It’s early. (adjective/predicate)

Yes, but it changes the meaning:

• Me desperté temprano = I woke up early (relative to normal).
• Me desperté por la mañana = I woke up in the morning (time of day, not necessarily early). You can combine them: Me desperté temprano por la mañana.

Y simply links two past events. You can swap it depending on the relationship:

• ... y me desperté temprano = and I woke up early (simple sequence/extra info)
• ..., así que me desperté temprano = so I woke up early (cause-effect)
• ..., pero me desperté temprano = but I woke up early (contrast)

Accent marks depend on stress rules and pronunciation:

• desperté needs an accent to mark stress on the final syllable (and to distinguish it from other forms like desperte which would follow default stress rules).
• anoche, tuve, un, sueño, raro, y, me, temprano follow normal stress/spelling rules without written accents (note sueño has ñ, which is a separate letter, not an accent mark).