1 - As mike says, primary returns would show just an indication of something at that location, no altitude or code. In the video you actually see the infringing aircraft with an SSR return (mode A 7000, a conspicuity code which broadly means 'not receiving an ATS', with no mode C data).
2 - You probably need to think about this from a different perspective. For a controller, amongst other differentiations, a flight is either 'known traffic' or 'unknown traffic'.
Known traffic means that the controller knows about a particular aircraft and what it is going to do. It may seem strange but it doesn't mean that the aircraft is talking to that controller, in some cases the controller may have been told that the aircraft squawking XXXX wants to fly through a particular piece of airspace that he/she is responsible for at 3000ft and has agreed to it. This is known as co-ordination.
Unknown traffic is simply a flight that the controller has no knowledge of. In a radar environment one example is the infringing aircraft, which the controller can see is there but it doesn't have a clearance. If the aircraft is showing a level that is above or below the airspace for which the controller is responsible then he/she can assume that it is outside the airspace unless there is any information to suggest that the level information is not reliable. Because the controller doesn't know if the level information has been checked, in this situation there is a margin for error built into the procedures.
In the UK the rules for controllers about what to do if there is unknown traffic in airspace that they are looking after depend on the class of airspace in question. In the video you were looking at the Heathrow area which is class A and the rules say ' If radar derived, or other information, indicates that an aircraft is making an unauthorised penetration of the airspace, is lost, or has experienced radio failure – avoiding action shall be given and traffic information shall be passed' - which is pretty much what you saw (the avoiding action bit, at least).
The comment - Because a mode C return can obviously be altered and be incorrect - I've many times flown with them being well off or have the wrong pressure altitude input - is based on an incorrect assumption I'm afraid. The level information contained in an SSR return is derived from a completely separate sensor from those the pilot is using and is always 'set' to 1013 hPa. It doesn't matter what setting the altimeter sub-scale is turned to, the mode C data will always show the level based on 1013 hPa. The radar display systems used by controllers will convert this information to show an altitude if this is necessary - if you do a search you'll find lots of posts on this particular topic and how it all works.
No doubt others will have typed faster than me but I hope this helps to explain things.