1: Itinerant Kingship of Early and High Medieval German Rulers
Charlemagne (d. 814) of the Carolingian Franks certainly had Aachen, and his namesake (after five centuries) Charles IV (d. 1378) of HRE had Prague as capital of his dominion (though as that of his own Kingdom of Bohemia). Do you have some ideas that which German city the rulers of Germany mainly resided in course of High Middle Ages, between these two Charles? They kept almost constantly travelled around their realm, from a palace to another palace, a royal estate to another estate, and also, a royal/ dynastic monastery to another monastery, with their household, or the whole ‘court’ with retinues. This was actually quite convenient way of ruling for the ruler who dominated relatively large-scale realm to communicate with the elites from diverse parts of the kingdom, only with very rudimentary apparatus of transportation or administration (Cf. Althoff 2005: 230-47). The subject used to have a chance to meet the ruler who travelled around in course of his itinerary near their own land, and also, to ask some favors in exchange for the dues (often as the supply to the royal entourage) (Reuter 1991: 208-13). In short, many German rulers, especially Ottonian and Salian (ca. 935-1125) ones, were probably not so familiar with the concept of the capital itself, as OP asked.
2: Who was/ were the closest counterpart?
Except for the coronation, medieval German rulers (HRE Emperor/ king) generally did not show much interest in City Rome itself. To give a famous example, Charlemagne himself in person left for Francia in 801 after the coronation and did not return to Rome/ Italy until his eventual death in 814. They usually put more emphasis on their lands north of Alps (as their original power bases), or the rich cities in Northern Italy (as, so to speak, their additional stack money purse). I can point out some Ottonian German rulers (919/36-1024), however, like Otto II (r. 973-983) and Otto III (HRE r. 996-1002), as somewhat exceptions to this general trend. Otto II got married with Theophanu, the Byzantine princess (Otto III was their son), and their policy as well as their representation of rule were strongly aware of the contemporary Byzantine empire that was still influential in principalities in Southern Italy.
Otto III was truly an exceptional figure for the history of City Rome in the Middle Ages. He held up policy of ‘Renovatio imperii Romanorum (The Renewal of the Empire of the Romans)’, occupied Rome with his army between 998-1001, set up his center of ruling at the new palace on the Palatino in Rome, and tried to put his relative/ close associate as Popes (Wickham 2015: 25-28): This was also a turning point for the papal history in a sense that the emperor chose a pope from non-Italian personnels at the first time. The second pope Otto III appointed as Sylvester II was his ex-mentor, Gerbert of Aurillac, famous scholar who had born as a son of commoner (farmer?) in Auvergne, France. Both Otto’s policy of Rome-centered ended in miserable failure, however. While the army of the Muslims crushed the ambition of the father, Otto II in South, the enemy of the son, Otto III was in Rome itself. The populace of Rome revolted against the emperor and his pope, and forced them to retreat from the city. City Rome and her mastermind did not always like excessive intervening emperor and his protégé.
It was Roman city aristocratic families like the family of Teofilatto, the Crescensians and the Tusclans, that kept real power in Rome at least the middle of the 11th century, and the pope were more than often just a puppet for their power struggle. While the origin of the Papal State is said to date back to the middle of the 8th century, it was not until the pontificate of famous Innocent III (r. 1198-1216) that the pope firmly established his control over it as a secular territorial lord with help of rectors, usually selected from cardinals or members of the papal court (Schimmelpfennig 1992: 171f.). In short, in addition to the pope himself, the city itself and its aristocrats often welcome neither HRE emperor nor the pope as the ruler, and their political influence could not be underestimated from the 10th to the 12th century.
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[Edited]: fixes typo.