"Please call me a taxi."
I agree. This sentence should just be "call a taxi" and the (for me) is implied.
The verb 下さる shares the meaning with くれる of "please do this thing for me or someone close to me" specifically. You can use 呼んでもらえませんか or the similar but more polite 呼んでいただけませんか but it's somewhat more formal.
I agree. If they wanted to say please call me a taxi it would be better to say タクシーを呼んでくれませんか。
You don't really use くれる and ください together like that. Instead you would say something more like タクシーを呼んでくれませんか。
Ahh I see. Thank you! I knew it works technically, but definitely くれませんか is better.
Is there an option to make this sentence more polite so it doesn't sound like a demand?
くれますか/くれませんか are very similar in grammatical function to their English counterparts (will you call me a taxi,please?/won't you please call me a taxi?) If you feel like softening it a little.