Why is the “的” necessary?
No, it's optional. 的 CAN be used when talking about the time duration of transitive activities ("We spent 30 minutes in the taxi", "we studied all day yesterday"), but it's not necessary to get the meaning across.
Is 我们三十分钟坐了出租车去宾馆 incorrect? Could someone explain?
Yes, that sounds wrong. You need the 三十分钟 (30mins) after the verb 坐了 (sat). Like in English, "we spent 30mins..." and not *we 30mins spent.
I always thought of it as how long goes between the verb and the object while the days and the date go directly after the subject or directly before
These questions where you have to get a long exact way to say it right are just so frustrating.
Can't I also say: 我们做了三十分钟出租车到宾馆。
I might very well be wrong, but it could be that you used 做 rather than 坐.
I said that and it marked it as right
What am I missing with "我们坐了出租车三十分钟去酒店."
Is 钟 really needed here?
I wonder, why is there the ''个'' in ''一个小时'' but is there not the ''个'' in ''三十分钟'' ? Is the ''个'' optional here?