What would you have me do?とWhat do you expect me to do?, What do you want me to do?はどこがどう違うのか？一見似たような意味に思えますが、言わんとしていることにはかなり大きな違いが存在します。話し手の置かれた状況を考慮する必要があるようです。
What would you have me do
What would you have me do for you?を直訳すれば、「君は僕にどうして欲しいというのか？」という意味になりますが、このイディオムに込められた話者の気持ちが問題になります。
“What would you have me do” vs. “What do you expect me to do
The tone of the conversation is the key to the meaning and why the author wrote as he did:
会話の調子が、その意味するところと書き手が何故そう書いたのかを知る鍵になります。
“What would you have me do?” = there seems to have been nothing else I could have done given the circumstances. Do you have a suggestion because the only suggestion I can offer is/will be an impractical one.
“What do you want me to do?” = What is your suggested action?
“What would you have me do?” =状況を考慮した場合、私がやれたこと以外のことは何もできなかったように思われる。非現実的な手段以外に手はないが、君の考えは？
“What do you want me to do?” = どうしてもらいたい？
A: “There is s storm in mid-Atlantic and our ships are sinking! The rescue aircraft and ships cannot get there!”
B: “What do you want me to do?” = How can I help, do you have a suggestion?
A: “Get the weather forecast!”
A:中部大西洋の嵐で、船が沈没中！救助艇と救助船は、そこには行けません！
B:私にどうして欲しいのですか？（私にできることがあったら言ってください）
A:天気予報を教えて！
A: “There is a storm in mid-Atlantic and our ships are sinking! The weather means rescue aircraft and ships cannot get there!”
B: “What would you have me do? [Stop the storm?]”
A: 中部大西洋の嵐で船が沈没中！悪天候のせいで救難機や救助船は動けず！
B: どうせいちゅうの？（嵐を止めろとでも？）
What do you want me to do?やWhat do you expect me to doは、何かやってもらいたいことがあったら言ってくれみたいな意味合いで、What would you have me doは、もう俺には何もやれることはない！的な意味合いになると、１人の米語ネイティブが言っています。
What do you want me to do?
In these sentences, I hear “of” and “from” as synonymous, with the meanings given by PaulQ, and with “of” sounding old-fashioned (I would never say it).
“What do you want me to do?”
これらの文で、what do you want of me?とwhat do you want from me?は、同義語と理解しています。意味はPaulQが言った通りで、what do you want of meの方は、かなり古臭く聞こえるので、私は絶対に使いません。”What do you want me to do?”と同義です。
What do you want me to do? = what do you want from me?と考えれば良いと、１人の米語ネイティブが言っています。どうして欲しいの？と覚えておけばいいでしょう。
What do you expect me to do
What do you expect me to do? = What do you want from me? = What do you want me to do?の公式が成り立ちます。What do you want with me?とはニュアンスがかなり違うようです。What do you want with me?は、何の用？(俺になんか用あんの？)、俺をどうしようというんだ(俺をどうしたいんだ)的な意味合いがあります。
What made you bring me here? what do you want with me?
どうして俺をここに連れてきた？俺をどうしようってんだ？
参考サイトwhat do you want with me? from me?