In the Star Wars novel The Truce at Bakura, Luke Skywalker is able to extract a deadly parasite he was poisoned with with from his lungs using the Force. In the Clone Wars cartoon series, several of the Jedi Masters are able to dismantle droids and use their nuts and bolts to rip through them.
A Jedi cannot manipulate an android's mind per se, but if you follow the canon of story where they are insanely powerful and able to manipulate things on a small scale using the Force, then it could be reasoned that a Jedi with the knowledge of building and programming droids (e.g. Anakin Skywalker), could reprogram a droid using the Force; effectively altering/controlling its mind.
Yes. In the (canon) junior novelisation for Return of the Jedi, we see Luke tricking the TT-8L gate guard droid with mind powers.
The way in turns out to be easy enough. An electronic eyeball barely has time to pop out before Luke has said, “You will open the door.”
Yes, this is a Jedi mind trick and it works easily. Inside, the simpleminded guard who operates the door unthinkingly obeys.
Droids dont have a conscious mind. I think its common knowledge that robots have computers, not brains. It is shown throught the series that ind tricks only work on the weak MINDED. This is shown when Luke attempts a mind trick on Jabba The Hut. You could see an example prooving this too when Ben Kenobi tricked the Stormtrooper using the old favorite quote: "These are not the droids you are looking for."
In summary: Mind tricks only work on weak minded, droids do not have inds therefore a mind trick would be useless against a droid.
Legends: Yes, by another name
In Legends, a Force ability was known by the name of "mechu-deru" that allowed the user to not only influence droids, but to control or reprogram them entirely. From a now-deleted Hyperspace article on starwars.com:
It is believed that through their bizarre alchemies, the Sith somehow stumbled upon the ability to manipulate mechanicals of complex nature through the Force: ships, computers, and, indeed, even droids.
There seems to be a range of ways that mechu-deru could interface with droids and mechanical objects, from simply allowing the user to understand mechanical systems better (as described in The Jedi Path) to turning living beings into cyborgs called "technobeasts" (known as mechu-deru vitae). However, the ability definitely included control over droids:
Because droids are not organic beings, they were a subject few Jedi bothered to understand. Jedi powers that had a technical influence were usually crude, such as the Force ability mechu macture, or, "destroy droid." The Sith went farther than the Jedi ever did, developing the ability mechu-deru to allow for Force control over mechanical systems. In true Sith fashion, the power stressed absolute control, turning droids into puppets.
(source: The New Essential Guide to Droids (2006), page 195)