What is the nature of God's wrath upon those who do not supplicate Him, and are they punished for that? Is it considered a sin that warrants entry into Hellfire? And how can one increase their supplication? What is the ruling on dedicating supplication solely to asking for Paradise?

In summary:

The hadith "Indeed, whoever does not ask Allah, He becomes angry with him" has been deemed hasan (good) by some scholars and da'if (weak) by others. However, assuming its authenticity, it indicates that not asking Allah constitutes arrogance and self-sufficiency from Him, which is impermissible for a servant. Allah loves to be asked. And whoever incurs Allah's wrath, He will punish them unless He pardons them.

One should frequently supplicate and be persistent in it, asking Allah for everything concerning matters of this world and the Hereafter, for Allah loves those who are persistent in supplication, and nothing is too great for Him. It is also mentioned in a Hadith Qudsi that Allah Almighty answers His servants when they ask Him, and that does not diminish anything from what He possesses.