Can God for example sigh? Can He grieve? Can He burn with anger? All of these examples in some form have Bible backing.
Or are these merely metonymic figures of speech, metaphors?
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The answer is YES, God is capable of human emotions.
The common topic in theology for this question is the discussion on 2 contrasting attributes of God: passibility vs. impassibility. This answer borrows heavily from the two GotQuestions.org articles linked above.
Let's consider the data:
As you mention, Scriptures presents God in emotional terms, suggesting His passibility. Another clear data is how all mainstream Christianity teaches that Jesus is fully human, and thus should be capable of the full range of human emotion, including his fear, anger, sadness as clearly depicted in the gospels.
Allowing this, another clear teaching is that those desires of God and Jesus are well-ordered: proper and without sin, like what humans should have.
Another clear teaching is from the angle that everything God created must have originated from God, including emotions. Protestant theologians agree with St. Anselm's famous ontological argument for God: "nothing can be greater than a being than which no greater can be conceived". Because we experience emotion, God must be capable of that too.
But this creates an apparent conflict with another attribute that we would like God to have: impassibility. Not impassibility in the sense of being aloof, untouched, apathetic, but impassibility to support His immutability (unchanging nature). So we want Him to grieve, but not "crippled with grief". We want Him to support our freedom of choice, but not catch Him by surprise (like in Open Theism).
It's a delicate balance, and the Scriptures seems to chart a middle ground, showing how:
Although His nature and character remain the same, the way He responds to us may differ according to the situation. He always responds in ways that are consistent with His other attributes.
Jesus is key to proper balance as the article concludes:
perhaps in Christ we can understand immutability in a way that will allow us to see a perfect, unchangeable God who still experiences pain and emotions. Hebrews 13:8 says, “Jesus Christ is the same yesterday and today and forever,” yet we know that Jesus changed—He grew, He acted, He had emotions, He died, He rose, and He felt anger, compassion, and hunger—yet His changes never indicated that He was anything less than perfect to begin with.
Another way of conceiving God as capable of human emotions is to consider how the 3 persons in the Trinity love one another and is linked to joy, allowing us to realistically participates in the Joy of God, a supreme emotion. John Piper (a Reformed Protestant pastor and theologian) explains this vision of God (from Jonathan Edwards) in his 2020 article Can We Explain the Trinity? My Favorite Image for the Greatest Mystery. There is a major section of the article ("Words: Inadequate and Indispensable") that help us to understand the proper role of "metonymic figures of speech and metaphors" you mention in your question to understand God.