From the videos it is not possible to determine the exact brand/model of mic that is being used, but some general information.
Today in live theatre virtually all of the main characters are miced with Shure, Sennheiser and AKG units being among the most popular.
In the theatre with which I am associated, we use 48 channels (ie 48 separate wireless mic units) all being Sennheiser belt pack transmitters and for the microphone we use the DPA units (the mics alone are about $Aus500 - $600 each and come in a range of colours (light beige for Anglo-Saxon type skin, Black for those with dark skin and a slightly yellow colour for Asian type skin). The actual microphones are about the same size as a match-head so from about a few feet away they are virtually impossible to see (if mounted correctly). AND when stepped on can not be repaired, so a reasonable proportion of our budget goes into mic replacement !!!!!!!!
For most stage performances (eg Phantom, Les Mis, etc) the belt pack is generally placed in a small carry bag (either purchased or costume department made) and is then fitted (placed) in the most suitable location (quite often placed at the back of the actor and often down inside their pants --- the bag being washed after every performance as necessary).
The cable is generally fed up the actor's back (with a female often under the bra strap), past the back of the neck and often fed through the hair to sit just above the hair line (hence the term --- "hair-line mic"). The cable being held in place as it passes through the hair by the use of "toupe clips).
For males if they have a beard/side burns/etc the cable is sometimes threaded through this with the mic being placed in the beard.
At times you have to get very creative in where to place the mic, for example in a recent production of My Fair Lady, all of the nuns had their mic glued to the underside of their habits. On another occasion in a female tap-dancing production, we had to run the mic cable down the inside of their stockings and glue the mic to their ankle so as to get a really clear tapping sound over the sound of the music (eg like the tapping sound you hear on productions like River Dance --- not saying that is specifically how they got the sound on those productions however).
Quite often with young children (especially if doing a song/dance routine), because of their very soft voices, with the use of VERY special "face-tape" we actually run the cable up their back, through the hair and over their ear, with the cable running across the face and the mic being placed about 1" from their mouth, but because of the smallness of the mic coupled with its colour and that of the minutely thin cable, from about 10ft the system is virtually impossible to see from the audience.
The most important part of the placing of the mic is the time taken to VERY carefully equalize EVERY microphone to the performer and to place the microphone in a position where it is not picking up stage noises or most importantly other actors who might be close by, especially if they have no mic.
To achieve the best results generally requires a LARGE financial investment in the complete receiver/transmitter/mic unit, However having said that, I have recently imported a system from China (not a knock-off but fully designed and engineered) that is in my opinion equal to the best, but at a cost that is quite amazing (probably less than 1/10th the cost of any of the better known brands).
Re connecting the receiver unit to a camera.
Depending upon the camera (ie straight video or DSLR) it might have an audio input, but unless a better quality semi/fully professional video camera (eg some of the Sony and Panasonic units), most will only have an unbalanced microphone input.
The camera that I have been using mostly for video clips, is the Nikon D-7000 (there are newer models) which has an input for a stereo unbalanced mic (-50db), but my wireless mics (an 8-channel unit) and music backing track playback units are all "Line Level" outputs (either +4db or -10db), so I put these through a mixer and plug the stereo output from the mixer (+4db) into a small two channel +4db to -50db pad unit I designed and this is connected to the camera's mic input socket. I also send the backing track to a small portable PA unit for the artist(s) to hear and sing to.
By doing this I record full CD quality in perfect sync directly to the camera --- saves a lot of time and syncing effort later in the post production stage.
To simply use a condenser mic on a stand for the purpose that the originator requires, will not give a really good sound unless experienced microphone boom operators are employed and a high quality directional mic is used, but this type of mic will generally require phantom volts and a balanced output (ie a camera mic input that is balanced and can provide phantom volts to the mic has to be used). VERY few non-professional cameras will be able to have these facilities.
Although somewhat lengthy, I trust that there is some useful information contained within all of the above.
David