I think in universe a Jedi who would do that would be a potential dark-sider, or it would at least conflict with the Order's ideals. They seem very weary of politics in general, and they try to do their part while hoping the Senate will mostly handle itself. I'm just guessing, but at first they seem to keep an eye on politics yet not get directly involved. The Clone Wars and the return of the Sith arguably led them to compromise their values. I haven't followed EU though, so others will have a better answer.
The Jedi weren't mind-controlling the Senate. To begin with that wouldn't be very Jedi-like and more in line with the Sith. The Jedi worked with the Senate and in some sense, they worked for the Senate. To quote young Palpatine: Now I'm sure working with the Senate, the Jedi did a lot of good. The real problem is that the Senate was full of corruption. Which the Jedi should have investigated and dealt with. Also I think the Jedi had somewhat forgotten that they serve the Force, not the Senate.
An..interesting question. I see this question as the natural corollary of the 'TCW mindset', the possibly EU-lead construct that the Jedi are 'to blame' and are so because they are, in some deep way, 'corrupted'. Yjey are 'corrupted' on the basis that they somehow forgot that they serve The Force not the Senate - completely obliterating the word between those two which is Republic, which is what they are actually sworn to serve. I say it is an interesting question because it is a perfectly valid projection given those ideas (of a 'corrupt' Jedi Order, 'corrupt' on the religiously narrow notion of serving the Force which is in some ideological way incompatible with serving the Republic) I think it particularly interesting in terms of how I personally am more and more interested in the movies - in terms of the differences between the conceptions used within the structure of the different incarnations. I think this argues, actually, for the virtual obliteration of the original conception of the Jedi. One might even ask, what really is the difference between the Jedi Order and the Sith within the currently structured climate?
Please keep this thread within the realm of, say, 'why it would or wouldn't be right for them to do this - and why,' or 'this aspect of their film (and to a lesser extent "canon"/EU) personalities would make me think they would/could/wouldn't/couldn't do this.' Threads like this one have a tendency to rely very heavily on EU/non-film sources and to just discuss what happened according to them - which would fit in more to the Lit forum. That isn't a response to anyone in particular here (I actually am kind of interested in where only one kenobi is going); it's just a general thought.
It was part of the Ruusan Reformation that the Jedi renounced ruling as part of the political system and swore only to serve.
I generally refuse to blame the Jedi for their own downfall. The Jedi swore themselves to serve the legitimate government of the Galaxy for countless centuries. During which the Jedi provided security and acted as a stable influence on the Republic. Unfortunately those they swore to protect turned corrupt, greedy, and betrayed the Jedi when they needed help the most. While the Jedi from our perspective and with our behind the scenes knowledge may perhaps should have made different choices, the Jedi in my opinion should not be blamed. The Jedi should not be considered corrupt simply because they unknowingly supported the most evil and corrupted individual in the whole Galaxy: Paplatine.
The Jedi were very involved in politics, just not in a military role till the Clone War. They mainly served as advisers, ambassadors, bodyguards, delegates, police force, and security advisers for the senate or other planet leaders. They have their codes and regulations which forbid them to abuse their powers against politicians or manipulate them with the force.
They weren't involved in the politics of the Republic, they merely followed orders on its behalf, in order to keep the peace and maintain its integrity. This somewhat super-powered organisation wouldn't have been tolerated by the galactic populace otherwise. Qui-Gon & Obi-Wan's mission to Naboo in TPM was acknowledged to be a bit off-the-record - you could imagine that such missions occurred now and then, but the very fact that it was 'secret' indicates just how sensitive the involvement of the Jedi in such matters tended to be. One of the key elements of Palpatine's overthrow was the claim that the Jedi had become involved in a major way (which they did, purely out of necessity, and they were entirely justified in doing so when attempting to arrest Chancellor Palpatine/Darth Sidious), and, thus, had crossed a very sacred line - tapping into what was probably a very widespread fear, one that would easily overshadow any accusations regarding a darker order that had become the stuff of fairytales and urban legend. It'd be much the same situation if the FBI found definite proof that Barack Obama was the son of Dracula, but he had them all arrested for being Islamic terrorists first.
The Jedi had accepted quite some time ago, that they should only be the servants of the Republic. Not its rulers. But the disgruntled Jedi who disagreed with this, felt that it was their right to rule. As a result, these rogue Jedi were the ones who created the Sith Order and began their campaign to take control of the Republic by force and create an empire. Because the Jedi refused to rule, or to use the Force to influence the Chancellor or any senator, this is why Yoda said that grave care must be taken with Palpatine. While Mace was correct that they would need to have some control over the Senate, until a new Chancellor was elected, such thinking was dangerous and could fracture the Jedi Order. Especially with the war already taking its toll on them. Palpatine knew this and it is why he was able to convince Anakin that the Jedi were taking over and that they were full of crap when it came to their rules and the Code.