The idea wasn't dropped. It was stated that their testimony would be useless unless they had evidence that would allow the court to accept the existence of mind control as an accepted fact.
That's not how the courts work. The Judge has discretion as to wether or not alleged facts will be assumed to be true by the court and will be allowed to be presented to a jury in an argument, based on various standards, including standards of scientific consensus and evidence. In our world, a defense attorney can't attempt to create reasonable doubt by arguing that their client was abducted by aliens, because there is no scientific consensus that alien abduction is a real phenomena. A judge would never allow the attorney to make that argument, nor would a judge in the MCU allow an attorney to argue that their client is innocent for reasons of mind control, because mind control is not something that has been established to actually exist in the MCU from a legal or scientific perspective.
Again, no, they didn't. It was explicitly stated that their goal was to force Killgrave to use his powers and to film it, thus providing video evidence of the existence of mind control. The idea of a forced confession was even brought up and dismissed on the grounds that you mentioned, because that's not what they were trying to do.
Also, remember that their primary goal in getting this evidence isn't to prosecute Killgrave, it's to create reasonable doubt for Hope's case. They want to establish mind control as a thing that exists and can be brought up as an argument in court so that Hope can get acquitted. While using illegal means to establish credibility for an alleged scientific fact is an unprecedented legal grey area and would stand a chance of tainting, or at least seriously overcomplicating, any attempt to prosecute Killgrave, that's also not their primary concern at the moment.