I can answer to the terrible. In Russian, he is known as Иван Грозный (Ivan Grozny). Grozny does mean terrible, but not in the sense of really bad. More is the sense of terror- and awe- inspiring. Think “I am Oz, the great and terrible.” I’ve seen it also translated as fearsome or even formidable.

(Incidentally, that is also why the capital of Chechnya is called Grozny. It was originally a fort and part of an effort to take control of the region by Russia in the 19th century, an effort that including ethnic cleaning and quite a bit of deliberate terror-inspiring, as reflected in the name of the fort).

Why Ivan would be considered terror-inspiring/fearsome/formidable, or even great and terrible is clear. He started off as an abused child puppet by powerful nobles. He ended up doubling Russia’s territory and building himself a central authority and powerful throne, the first Tsar of all the Russia’s, and he did it with the blood of many.

During his reign, he broke away from the Mongols/Tatars, centralised power at the expense of the nobles, violently subordinated previously independent city states and regions, and conducted a campaign of violence in support of the latter.

He is said to killed his eldest son in a fit of rage, he saw poisoners and plotters everywhere, and formed the infamous oprichnina, a private army who wore all black and had on their horses dog’s heads and a brooms, as a symbol that they would bite and then sweep all opposition away (he also executed the head of the oprochina). He had a temper too, and stories about him often involve a murder and vicious punishment. He lost the Livonian War after 25 years, and caused famine and depopulation in doing so.

He was also a learned man, with a library considered now a lost treasure, and for which treasure-hunters sometimes still seek. He introduced and codified new laws, including, for the first time, the punishment of imprisonment as an alternative to corporal punishment.

Ivan even looked the part - reportedly tall and strong (In 1963, archaeologists opened his grave and found he was about 178 cm talk and weighed 85 to 90 kg). To add to the image, he carried a heavy staff that Daniel Prinz von Buchau, an envoy of the Holy Roman Empire to Ivan IV’s court, said symbolised the tsar’s “imposing manhood.”

Overall, the kind of person who would make an impression, but also terrify.

As for when he got the name, that is not entirely clear. One version says that, on his birthday, a great thunderstorm struck Moscow, the likes of which had not been seen for many years. The storm was associated with his fearsome majesty. There do not appear to be any sources confirming this (there are very few primary sources at all for his reign, as compared to his stature within history now).

Another version says that Russian people began calling him the fearsome after his death, as a compliment, in respect to his strength and power, as compared to the weak rulers, wars and the so-called “Time of Troubles” that followed his death.

The third version is that he acquired the name informally while alive and it was reenforced and spread after his death, but mostly as a reflection of the deaths he caused and as a negative name.