This applies to ASA rules- other associations may have different appeal procedures.
An appeal can be made either during a live ball or a dead ball.
If the ball is live: The appeal may be made by any fielder who has possession of the ball by touching the base left too soon on a caught fly ball or the base missed, or by tagging the runner who committed the violation (if they are still on the playing field).
If the ball is dead: The appeal may be made by any infielder with or without the ball by simply making a verbal appeal to the umpire. On a dead ball appeal, there's no need to have the pitcher first get on the pitcher's plate, or wait until the ball is made live again, or throw to a base, or touch a base.
It is usually to your advantage to make a dead ball appeal if the runners have stopped advancing. Either request time- and wait until it is granted- or wait until the umpire calls time once the play is ended, then make the dead ball appeal. This eliminates the possibility of runners advancing while the appeal is being made, or even having a throw go out of play as you throw to a base, thus giving the runners extra bases on any base award. Of course, if the ball is still live and the runner is scrambling back to a base as the continuation of the play, you will need to make a live ball appeal before the runner gets back.
Most players- and, unfortunately, some umpires- don't understand the proper appeal process and that's when you get the yells to "give it to the pitcher, then throw it to the base!" which is all totally unecessary for a dead ball appeal.
On the other play you mention- the close play at the plate- if the runner misses the plate and the catcher misses the tag, the correct procedure is for the umpire to signal "safe". And the runner is safe- up until the point that he is properly appealed. "Doing nothing" tips off the defense that the base was missed and that gives them an unintended advatage in recognizing that an appeal is available to them. You want the defense to recognize the missed base on their own- without any unintended help from the umpire.