Hava çok soğuk, öte yandan göl kenarında kitap okumak istiyorum.

Yes. Both ama (“but”) and fakat (“however”) express contrast. You could say:
Hava çok soğuk, ama göl kenarında kitap okumak istiyorum.
Hava çok soğuk, fakat göl kenarında kitap okumak istiyorum.
Note that ama is more colloquial, fakat more formal, and neither carries quite the “on the other hand” nuance that öte yandan does. You also have alternatives like diğer yandan or öte taraftan.

Turkish often forms noun–noun compounds without a genitive suffix. göl kenarı literally “lake shore” is a compound. When you need a case ending (here the locative “at”), you attach it to the second word: göl kenarında = “at the lake shore.”
Alternatively, you could use the genitive-possessive construction gölün kenarında, which is equally correct:
– göl kenarında (compound + locative)
– gölün kenarında (genitive + “shore” + locative)

The suffix -da/-de marks the locative case (“in/at/on”). Because kenar ends in a back vowel a, vowel harmony turns -da into -nda (actually -nda is the assimilated form after a vowel). So:
kenar + - ında → kenarında (“at the edge/shore”).

kitap okumak istiyorum means “I want to read a book.” To express desire for an action in Turkish, you use the infinitive (verb + -mak/-mek) plus istemek (“to want”):
– okumak istiyorum (“I want to read”)
By contrast, okurum is the simple present (habitual) or sometimes future tense: “I read” or “I will read,” but it doesn’t express “want.”

Both are possible.
– kitap okumak istiyorum (infinitive without accusative) is very common in spoken Turkish.
– kitap okumayı istiyorum (infinitive + accusative -ı) is a bit more formal or precise.
Neither version changes the meaning: both convey “I want to read a book.”

hava çok soğuk (“the weather is very cold”) is a complete sentence with hava as subject, çok as intensifier, and soğuk as predicate adjective.
çok soğuk hava is just a noun phrase (“very cold weather”). To make a full sentence you’d need a verb, e.g. çok soğuk hava var (“there is very cold weather”), which sounds odd. So for simple weather statements, use hava + adjective.

Yes. You can front it for emphasis:
Öte yandan, göl kenarında kitap okumak istiyorum.
Turkish has fairly flexible word order, especially with adverbial connectors like öte yandan.