Learn Dutch: Ik denk het uitje gaat door, want het weer is mooi. - I think the outing is going ahead, because the weather is beautiful.

Because we dropped the word dat (that). Normally, you would say Ik denk dat het uitje doorgaat, where dat creates a subordinate clause and pushes the verbs to the end. In spoken Dutch, when you drop dat after a verb of thinking like denk, the following clause reverts to standard main-clause word order. So, the separable verb doorgaan splits: the finite verb (gaat) takes the second position, and the prefix (door) goes to the end of that clause.

Can I keep the word dat but use this same word order (Ik denk dat het uitje gaat door)?

No, you cannot mix and match. If you include dat, it forces the verbs to the end: Ik denk dat het uitje doorgaat. If you drop dat, you must use the standard main-clause word order: Ik denk het uitje gaat door.

Why doesn't the word want (because) push the verb is to the end of the sentence?

Because want is a coordinating conjunction, which means it simply connects two independent main clauses. Both clauses keep standard word order. If you used omdat, which also means 'because' but is a subordinating conjunction, it would push the verb to the end: ...omdat het weer mooi is.

How does doorgaan translate to 'going ahead' for an event?