Before the Clone Wars the Jedi were essentially hired by the Republic to serve as peacekeepers, and were supplemented in this role by the Judicials. When the Clone Wars started the Jedi served less as "peacekeepers" (since there was no peace) and instead as commanders of the Grand Army of the Republic and the Republic Navy.
Presumably, the Jedi were paid by the Republic for their peacekeeping and war-fighting services (and the Republic raised funds to pay the Jedi via taxation).
The Jedi Order appears to be (mostly) independent of the Senate according to this quote from Episode III:
PALPATINE: Anakin ... I'm appointing you to be my personal representative on the Jedi Council.
ANAKIN: Me? A Master? I am overwhelmed, sir, but the Council elects its own members. They will never accept this.
That said, the Jedi are likely subject to the Republic's laws just as any other Republic citizen. It is difficult to find examples of Jedi put on trial, but Ahsoka Tano was tried by the Republic for sedition (this is not conclusive, however, because she was expelled from the Jedi Order just before the trial). The Jedi are, of course, additionally subject to the rulings of the Jedi Council.
The Jedi are probably fairly well known, as even people on distant Tatooine are aware of them and their abilities. For example, from Episode I:
WATTO: What you think you're some kinda Jedi, waving your
hand around like that?
ANAKIN: Well, ahhh...you're a Jedi Knight, aren't you?
QUI-GON: What makes you think that?
ANAKIN : I saw your laser sword. Only Jedi carry that kind of weapon.
If minor individuals (such as a junk dealer and his slave on an Outer Rim planet like Tatooine) are aware of the Jedi then it's likely that the Jedi were fairly well known in the last years of the Republic. The Jedi gained even more publicity during the Clone Wars. The galaxy's apparent amnesia about the Jedi and the Force in Episode IV is mainly due to out-of-universe reasons (the Clone Wars were initially going to occur further in the past), but can be explained in-universe in part due to the fact that the Jedi were discredited and nearly eliminated at the end of the Clone Wars -- there were virtually no Force users left in the galaxy during Episode IV, and people apparently forgot about them pretty quickly (probably with some help from the Empire).