Can someone sit during his Shul's hakofos or is there an actual obligation to walk around the bimah? Does one have to hold the sefer torah? Does one have to say the pesukim in the Siddur that introduce each hakafah?
See Mishna Berura 669 sk 11:
...ולכן יש להתאמץ בזה לרקד ולזמר לכבוד התורה כמו שכתוב גבי דהע"ה מפזז ומכרכר בכל עוז לפני ד' וכ"כ משם האר"י ז"ל
והעידו על האר"י ז"ל שאמר שהמעלה העליונה שהשיג באה לו ע"י שהיה משמח בכל עוז בשמחה של מצוה וגם על הגר"א ז"ל כתבו שהיה מרקד לפני הס"ת בכל כחו: ...
The Mishna Berura mentioned a couple of responsa from MHRI Kolon advising to maintain all the customs. He concluded that we must make a special effort to dance for the prestige of Tora, and reported in name of the ARI zal this.
Additionally scholars testified that the Ari said about himself that his reached high degree the great gain from Simcha of Mitsva, and that the GRA was dancing with all his strength before t Sefer Tora.
Sitting may be a problem when people pass near you with the ST. Stand up may be obligatory. Solutions exist to permit sitting.
Can someone sit during his Shul's hakofos
No, see this link for few explanations.
is there an actual obligation to walk around the bimah?
Not an obligation but a strong advice.
Does one have to hold the sefer torah?
Not mentioned in SA Baer Heytev and MB. But there is mention of multiplying Aliot (RMA). (In many synagogues every body receive an Alya, but RMA did not say "everybody").
Does one have to say the pesukim in the Siddur that introduce each hakafah?
Not mentioned. Obviously it is impossible to call this an obligation. But maybe a problem to be "batel" when others are saying them. So if you are learning Tora no problem.