Yes. Explanation (with Verses for each claim made):

1) But didn’t God Love Jacob and hate Esau (Romans 9:13, Malachi 1:3)?

How about this Verse too which implies that God hates the wicked?

“God judgeth the righteous, and God is angry with the wicked every day.” (Psalm 7:11, KJV)

Answer: God hates for a time according to their sins but Loves eternally. Verse:

“A time to love and a time to hate…” (Ecclesiastes 3:8, NASB)

His hate is NOT eternal (as the Verse below implies directly toward ‘all the spirit’ He has made – in ‘all generality’, context):

“For I will not contend forever, Nor will I always be angry; For the spirit would grow faint before Me, And the breath of those whom I have made.” (Isaiah 57:16, NASB)

Yes, He does NOT cast any of the “sons of men” to be “forever” be separated from Him either (as the Consecutive Verses below prove in ‘all generality’ too – context of the phrase “sons of men”):

“For the Lord will not reject forever, For if He causes grief, then He will have compassion according to His abundant lovingkindnes. For He does not afflict willingly or grieve the sons of men.” (Lamentations 3:31 – 33, NASB)

But does it apply to the context of ‘afterlife banishment in Judgment’ (after they die)? Yes as the ancient Verse below means (implies) this too:

“We will certainly die and be like water poured out on the ground, which can’t be recovered. But God would not take away a life; he would devise plans so that the one banished from him does not remain banished.” (2 Samuel 14:14, CSB)

Is there an actual Biblical Example for this?

Yes, the “(formerly) disobedient spirits in prison” from the “days (time) of Noah” (1 Peter 3:18 – 19) were Visited in the afterlife punishment regions by Lord Jesus Christ Who Preached (hence past tense in Greek – an occured act) the Gospel to them thereby though “these may be judged in the flesh as men” (in those afterlife regions), they may eventually “live to God in spirit” too (post-Judgment, as implied in 1 Peter 4:5 – 6).

2) God hates the sin but Loves the sinner

God Loves the World (kosmos – all creation in it, John 3:16) but hates the ‘sin in the world’ (1 John 2:15 – 17).

3) How does it prove that God Loves Esau or Judas too and that His Hate for Esau cannot be eternal?

Verses:

“You have heard that it was said, ‘YOU SHALL LOVE YOUR NEIGHBOR and HATE YOUR ENEMY.’ But I say to you, LOVE [Agape] YOUR ENEMIES … Therefore you are to be PERFECT, as your Heavenly Father is PERFECT.” – Lord Jesus Christ (Matthew 5:43, 44, 48, NASB)

Can you see it?

In other words, the Old Testament Verses of “hate your enemy type” (which ‘was heard’ previously, e. g. “Esau I hated” etc.) —-> does NOT hold anymore as the TIME to LOVE [Agape] them has come (as Christ Cancels the Hate as evidenced by the phrase “But I tell you”) —-> making ONLY LOVE toward HIS ENEMIES too to remain ‘past the hate of Judgment’ —–> as this doesn’t just apply to men or us but is the PERFECTION of GOD the FATHER Himself as Lord Jesus Christ concludes in this Topic in Consecutive Verses above (Irrefutable Context) as it is Written.

In short, since His enemies (includes both Esau and Judas too) —> God Loves His enemies (as Matthew 5:44 with Matthew 5:48 imply, in Context) —> means that God Loves Esau and Judas too way past any required Judgment as God the Father is such Perfectly.

Conclusion

This is the “Unfailing Love (or Agape) of God in Lord Jesus Christ” (1 Corinthians 13) which will SAVE any SINNER in the WORLD (Kosmos) eventually, to quote:

Yes, Lord Jesus Christ came to ‘save all sinners in the World’ (not just humans but any sinner in the ‘world’) as it is Written in verse below:

“stedfast is the word, and of all acceptation worthy, that Christ Jesus came to the world to save sinners — first of whom I am;” (1 Timothy 1:15, YLT)

Isn’t Lord Jesus Christ Truly Amazing?

Because He Saves All the World (John 3:17, 1 John 4:14) by taking away ‘All Sin in the World’ eventually (1 John 2:2, John 1:29) when His Ransom is applied to All eventually, becoming a Testimony or ‘comes to Pass’ in its own Divine Times (1 Timothy 2:6) —> let All Glory & Honour be to Him only!

P/S:

Can God’s Hate in Judgment Save?

Yes. How? Here’s a quote from the write up below for edification:

How about Hate?

God’s Hate toward evildoers (Psalm 5:5) cannot be understood via ‘human hate’ because even in Hate, God is Righteous (James 1:20).

How is God Righteous in Hate? Since ‘God is Love’ (1 John 4:8), even His Hate (e.g. “all His Burning Anger”, Zephaniah 3:8, NASB) —-> must have a Motive in Love, namely to cause them all —> upon whom such a Hate causes Judgment first where ‘these feel the pain & receive their own evil first’ [as described in point (iii) prior] —-> to ‘serve Him’ eventually and have ‘purified lips’, i.e. for their ‘purification too’ as mentioned in the next consecutive verse, Zephaniah 3:9 in perfect context.

Indeed:

“Therefore wait for Me,” declares the LORD,

“For the day when I rise up as a witness.

Indeed, My decision is to gather nations,

To assemble kingdoms,

To pour out on them My indignation,

All My burning anger;

For all the earth will be devoured

By the fire of My zeal.

“For then I will give to the peoples purified lips,

That all of them may call on the name of the LORD,

To serve Him shoulder to shoulder.” (Zephaniah 3:8 – 9, NASB)

The earlier chapters in Zephaniah 1 – 2 speak mainly of the “Judgment part” agreeing with point (iii) prior while the verses quoted here speaks of the “Mercy part” agreeing to James 2:13 perfectly too.

Are we able to understand it as it is Written?

To learn more about the ‘Fierce Anger of the LORD’ (Jeremiah 30:24) which Aims at achieving His Will to Save the World too, please consider the write up below:

https://www.anonymouschristian.org/blog/hell-hate-and-love/

Please note the words “visible and invisible” (creation, which certainly includes Hell, Hades or Sheol, the afterlife, Gehenna, the Lake of Fire, Satan or anything else) too in Colossians 1:16 which are mentioned (collectively as a whole which/who are all part of “all created things” in this ‘only’ existing Biblical Definition) —–> which is then Promised to be “Reconciled” back to “Him” (Lord Jesus Christ, God) in Colossians 1:20.

A Beautiful Irrefutable Truth from Scripture “as it is Written”.

Another post regarding this for further edification, please consider (thank you for reading!):

www.anonymouschristian.org/blog/power-in-the-blood-of-christ-nothing-compares-nothing-unsubjected-eventually/

Truly, God doesn’t Change from being LOVE!

“The one who does not love does not know God, for GOD IS LOVE. We love, because He first loved us.” – apostle John (1 John 4:8, 19, NASB)

All Glory to God Alone!

Here’s a Verse agreeing to all this Depth of Theology regarding His Perfect and Unfailing Love too:

“All [Created] things have been handed over to Me by My Father, and no one knows who the Son is except the Father, and who the Father is except the Son, and anyone to whom the Son wills to reveal Him.” – Lord Jesus Christ (Luke 10:22, NASB)

[Emphasis mine]

How does God doesn’t change?

“Once God Loves, He always Loves;” – Unknown

Since God Loved all creation in the World (John 3:16, 1 Timothy 1:15, 1 John 2:2), He cannot Stop Loving All Creation in the World till He Saves the World (1 John 4:14) causing Sin (corruption, decay) to cease to exist (Romans 8:20 – 21) when All Creation becomes BORN AGAIN (Romans 8:22) and He becomes ALL in ALL [Creation, 1 Corinthians 15:28].