I can't speak to the first part of your question, but even when Rome was relatively small, it was still extremely politically and religiously important - it's where the Pope lived (well...usually), and all Catholic bishops and abbots were supposed to visit the Pope to be officially appointed to their offices. Maybe less attention was paid to the old Roman sites, but it was also full of churches and cemeteries and other Christian pilgrimage sites. So it was always a very busy place, even if only 20,000 people lived there (and 20,000 people would still be a pretty big medieval city!).

“Rome was the richest of all the cities in the West in relics and shrines and thus the development of the cult of saints undoubtedly enhanced this city’s importance as a pilgrimage centre.”

(Debra J. Birch, Pilgrimage to Rome in the Middle Ages: Continuity and Change (Boydell, 1998), p. 26)

Visitors were mostly religious pilgrims on their way to the various churches associated with the early Catholic saints and martyrs. Most importantly, Sts. Peter and Paul were believed to be buried there. Today, St. Peter's Basilica in the Vatican is one of the biggest churches in the world, and the same was true back then, although it was a different building. The first basilica was built all the way back in the 4th century, and lasted until the new one was built in the 16th century. It was full of burial sites and statues and mosaics.

“The flow of pilgrims was also fuelled by the rapidity of the Muslim advance through the Near East in the seventh century, which made pilgrimage to the Holy Land increasingly difficult.”

(Birch, p. 38)

There are lots of accounts of pilgrims to Jerusalem, but even during the crusades when Jerusalem was briefly under Christian control again, it was far away and hard to get to. Visiting Rome was much easier and pilgrims sometimes left descriptions of what they saw there as well, such as the Notitia ecclesiarum urbis Romae (a "list of churches of the city of Rome") and De locis sanctis martyrum ("on the holy sites of the martyrs").

There are a couple of famous accounts by pilgrims who also visited ancient Roman sites. There is one by "Master Gregorius" (The Marvels of Rome, ed. and trans. John Osborne (Pontifical Institute for Mediaeval Studies, 1987)), who saw the the Forum, and the Colosseum. He saw art such as a statue of Venus, and a statue of the wolf suckling Romulus and Remus (although whether this was the same statue that is currently in the Capitoline Museums is the subject of debate...). Osborne's translation does not seem to be online anywhere, but if you can read the Latin, it's available here.

Another account is the Mirabilia Urbis Romae by Benedict of San Pietro (ed. and trans. Francis Morgan Nichols (London, 1889) - Gregory and Benedict's accounts have basically the same title, as this one also means "the marvels of Rome"). Benedict has a long section about the Pantheon, which he found very impressive, but it’s mostly just a list of things he saw, like the “memorial of the emperor Hadrian”, the “winding pillar of Trajan”, the “Colossean Amphitheatre”, etc. There aren't really many lengthy descriptions.

The editor notes that in one chapter

“the visitor is led from the south side of the Capitol, across the Roman Forum, and up the Sacred Way; but it should be observed that the names of these famous localities appear to have been forgotten.”

(Nichols, pg. 96, note 5)

So, certainly Rome was a very significant and popular site for pilgrims and tourists, and was definitely something to be marvelled at. They knew the locations and names of many ancient Roman sites, although some of them were obscure by then. Pilgrims were more concerned with the later Christian sites, which had sometimes been built overtop of the Roman sites or were located elsewhere in the city.