Ben dün gece kitap okurken kardeşim odasında piyano çalıyordu; o melodi beni sakinleştiriyordu.

In Turkish the subject pronoun is usually optional because the verb ending already shows who is acting. Ben (I) is added here for emphasis or clarity—perhaps to contrast your action with your brother’s. You can drop it without changing the meaning:
• Dün gece kitap okurken…

Çalıyordu is the past continuous form (“was playing”), built from the root çal- + progressive -ıyor + past -du.
• çalıyor = “is playing” (present continuous)
• çalacaktı = “was going to play” (future-in-the-past)
We use çalıyordu because the brother’s piano playing was ongoing at that past moment (“while I was reading”).

To specify “in his/her room,” Turkish uses a possessive suffix on the noun before the locative. The steps are:
1) oda (room)
2) + 3rd-person possessor -sı ⇒ odası (“his/her room”)
3) + locative -nda ⇒ odasında (“in his/her room”)
If you said kardeşim odada, it would mean “my brother was playing piano in the room,” but you wouldn’t explicitly mark whose room it is.

Sakinleştiriyordu = “was calming (me).” Breakdown:
• sakinleştir- = causative of sakinleş- (to become calm) ⇒ “make calm”
• -iyor = progressive aspect (“is/was …ing”)
• -du = past tense marker
Because ­yor + ­du contract to ­yordu, you get sakinleştiriyordu. The object beni (“me”) shows who is being calmed.

Yes, both can translate as “last night,” but:
• dün gece is more direct (“yesterday night”).
• geçen gece literally means “the night that passed” and can also mean “the other night” (a few nights ago).
In everyday speech, dün gece is the most unambiguous for “last night.”

Turkish punctuation allows semicolons to link closely related but distinct clauses. You could replace it with a comma:
• …, kardeşim odasında piyano çalıyordu, o melodi beni sakinleştiriyordu.
Using ve (“and”) would make it more of a simple list of actions, whereas the semicolon here gives each clause its own weight and shows a closer causal or descriptive link.