The 1203-4 siege and its aftereffects were by far more destructive. Constantinople in the 12th century was a metropolis of several hundred thousand people and a major center of political and economic power. Just prior to the sack, there was a devastating fire that leveled swaths of the city. The Crusaders were there because they hadn't been paid so there was a huge motivation to strip the city of everything of value. Most famously the Horses of St. Mark in Venice are four bronze sculptures dating possibly to antiquity that were looted from the Hippodrome, but many relics and riches were stolen and disbursed around Europe in the aftermath.

Byzantine Constantinople existed as it did by being the center of an efficient, centralized tax-collecting state. But after their initial successes, the Crusaders/Latin Empire rapidly declined and was soon itself largely confined to Constantinople. The Latin Empire instead splintered into many principalities nominally subject to the Emperor (The Kingdom of Thessalonica, numerous duchies in the Aegean and modern Greece), Byzantine successor states centered around Nicaea, Epirus, and Trebizond (itself an important trading hub), and lands cleaved off by the Bulgarians and other independent-minded former Byzantine subjects. Help from the west cost money, so the Latins continued to plunder the stored up wealth in the city. The last Latin Emperor to reign in Constantinople infamously sold the lead roof off the palace, leaving it largely uninhabitable by the time Michael VIII Palaiologos retook the city.

In comparison, the city in 1453 held maybe 50,000 inhabitants and contemporary 15th century accounts describe the city more resembling a collection of villages within the ancient walls. A Castillian ambassador traveling to the court of Timur in the early 1400s had this to say:

Though the city is so large, it is not all well peopled, for in the middle of it there are many enclosures, where there are corn fields, and fruit gardens. The most populous part is near the sea ; and the greatest traffic is from the city, by the gates which open on the sea, especially the gates leading to the city of Pera, on account of the ships which go there to unload ; and because those of both cities pass from the one to the other, with their merchandise.

This city of Constantinople contains many great churches and monasteries, but most of them are in ruins ; though it seems clear that, in former times, when the city was in its youth, it was the most renowned city in the world.

Do we know approximately how many people were killed in each case after the city had fallen?

Not really. Certainly the Crusaders and Ottoman troops engaged in murder and atrocities, but many more people would have the opportunity to flee, at least until things calmed down. After the 1453 siege, many captured notables did end up being executed but some escaped on Venetian or Genoese ships in the confusion and ended up in Italy.

What became of the Christian survivors in 1453? Did most leave the city, or did most remain?

Mehmed II was very much interested in rapidly revitalizing the city; his Ottoman Empire had by then occupied roughly the same space as the Byzantine Empire did at its greatest medieval extent under the 11th century and Constantinople was perfectly positioned as a central capital. After the traditional 3 day sacking period, he allowed the survivors to return if they wished under his protection and distributed money to them so they could ransom back their relatives. While he converted the Hagia Sophia into a mosque, he initially allowed the Christians to maintain the Church of the Holy Apostles, the second-largest church in the city. Now that the city was reestablished as the capital of a major empire, it rapidly recovered.

Of those who remained, did they largely remain Christian or convert to Islam, and if so, over what period of time?

This is getting out of my area of expertise, but Constantinople remained a cosmopolitan city up until the end of the Ottoman Empire. Turks were probably a bare majority and there were large Greek, Armenian, and Jewish communities.