eki de tomodati wo matte iru aida, kaisatuguti no tikaku wo urourosite imasita.

Because で marks the place where an action happens.

Here, the action is 友達を待っている (waiting for a friend), so 駅で means at the station as the location of that action.

• 駅で待つ = wait at the station
• 駅にいる = be at the station

So で is the natural choice here.

In Japanese, 待つ takes a direct object, so the person or thing being waited for is marked with を.

• 友達を待つ = wait for a friend
• バスを待つ = wait for the bus

Even though English uses for, Japanese does not here.

あいだ means during the time that... or while....

So:

• 待っている = is waiting / am waiting
• 待っているあいだ = while (someone) is waiting / during the time (someone) is waiting

In this sentence, it sets up the time frame for the main action: while waiting for a friend at the station...

This is a very common Japanese pattern.

The main verb うろうろしていました is in the past, so the whole scene is understood as past. The 待っている part describes an action that was ongoing at that time.

So Japanese often says:

• 待っているあいだ、うろうろしていました。 = While I was waiting, I was wandering around.

Even though いる looks present, the past meaning comes from the main clause and the overall context.

Because 待っている emphasizes the ongoing state of waiting.

待つ by itself is the plain dictionary form of to wait, but 待っている is more natural when talking about being in the middle of waiting.

So:

• 待つ = to wait
• 待っている = to be waiting

Since the sentence describes what happened during that ongoing period, 待っているあいだ sounds very natural.

Good question. The difference is often:

• あいだ = something continues throughout that period
• あいだに = something happens at some point during that period

Here, うろうろしていました is also an ongoing action, so あいだ fits well:

• 待っているあいだ、うろうろしていました。 = I was wandering around while I was waiting.

If the second action were a one-time event, あいだに might be more natural.

Usually yes, but with movement verbs, を can also mark the area someone moves through or around.

Here, うろうろする involves movement, so:

• 改札口の近くをうろうろする = wander around near the ticket gate

This を does not mean 改札口の近く is a direct object. It marks the place along which the movement happens.

You see the same kind of を in sentences like:

• 公園を歩く = walk through/in the park
• 道を走る = run along the road

うろうろする means something like:

• wander around
• pace around
• loiter around
• mill about

The nuance is usually that the movement is:

• a bit aimless
• repeated
• restless
• without a clear destination

So it is not just walking. It suggests moving around back and forth or hanging around the area.

うろうろしていました shows an action that was ongoing over a period of time in the past.

• うろうろしました = I wandered around
• うろうろしていました = I was wandering around / had been wandering around

Since the sentence describes what the person was doing during the waiting time, the progressive form していました is more natural.

Yes, something like 友達を待ちながら、改札口の近くをうろうろしていました is possible.

But there is a nuance difference:

• ながら = doing one action while doing another; the two actions are tightly linked
• あいだ = during the span of time when something is happening

So あいだ emphasizes the time period of waiting.
ながら would emphasize the simultaneous actions themselves.

Also, ながら normally requires the same subject for both actions, while あいだ is more flexible.

Because Japanese often leaves out the subject when it is clear from context.

In a sentence like this, the listener can naturally understand that the person waiting and wandering around is the speaker or the person being talked about.

So instead of saying:

• 私は駅で友達を待っているあいだ、...

Japanese often just says:

• 駅で友達を待っているあいだ、...

This sounds natural and not incomplete.

改札口 means ticket gate or ticket barrier/gate area.

• 改札 refers to the ticket-checking gate area more generally
• 改札口 literally means the gate entrance itself

In everyday Japanese, both can appear, and both are natural in many situations.
Here, 改札口の近く simply means near the ticket gate.