Apologies for the awkward phrasing but what I’m really asking is how similar was the Roman Senate to modern legislative bodies such as the US Congress or the British Parliament. Like did the Senate have committees in the way modern legislative bodies do? Did they have officers (ie a leader, a whip, majority leader etc.)? Did they have specific responsibilities and how were they chosen (like districts, or constituencies etc.)?
The Roman Senate was absolutely not a legislative assembly. They were a body of advisors who offered advice to magistrates. They often advised on legislation the magistrate might present to the popular assembly or the centuriate assembly, which both WERE legislative assemblies, but they could not make law.
The best illustration of this I know is the Lex de imperio Vespasiana. Vespasian didn't consolidate his power through an act of the Senate; he took his motion to the Popular Assembly, because the Senate could only offer an opinion, but not make law.
In practice, magistrates would often not bring a proposed law to the popular assembly if the Senate had not issued an opinion supporting it. The Gracchi and Caesar both violated this custom, with different effect. The Senate lynched both the Gracchi while Caesar.... well, was Caesar.
As for officers, meetings were officiated by a number of magistrates. Usually consuls; if I remember correctly praetors and tribunes could also officiate. There was a set order of speaking; princeps senatus (most senior member) spoke first, followed by the consuls elect, and then down the line of other magistrates elect (magistrates in office were not technically members of the Senate).
The presiding magistrate could raise or table motions pretty much at will, like a certain turtle in our Senate has done recently. Notably, during Caesar's war against Pompey, Caesar proposed a peace agreement that passed overwhelmingly, but the consul Marcellus rejected the vote with the support of 20-something senators out of 300.
While a senatus consultum (opinion of the Senate) wasn't law, it could be used as legal precedent. Cicero used a senatus consultum to execute the conspirators aligned with Catiline in 63 BCE. Clodius prosecuted Cicero for that execution based on a lex, which prosecution forced Cicero to flee the city, so you can see the relative strength of the two, senatus consultum and lex. Cicero was later recalled though.
TLDR: The Senate had parliamentary procedure, but did not make law, but its opinions carried a great deal of weight with magistrates who sometimes relied on them for justification of controversial actions.