Funny, I'm currently planning my first cabinet using a 21" Dell CRT with a Trinitron tube.
As for the MDF, I don't think you should have any issues providing the support base is properly reinforced and you are using 16 mm thick panels. MDF is rutinely used for furniture over Europe; it's way more resistant than it looks and it will last pretty much forever as long as you don't disassemble it. Make sure to use some decent wooden planks (pine should work nicely) so you can screw the monitor's MDF shelve to it in the proper angle using some timber connectors. If you are still unsure about the capacity of your MDF shelve to hold the monitor (it shouldn't be a problem, really), try using some extra beam/s running down to the base of the cabinet itself for extra support, either screwed to the sides or in central position. Of course, the monitor should have some proper support at the back so it doesn't slide over if you plan to set it at an angle.
Either way, I wouldn't recommend decasing your monitor in full. It's really dangerous and probably not necessary at all. Unlike TVs, which are often protected by HUGE plastic cases, computer monitors are relatively tightly fit and there's little space to gain from leaving the tube in the open if you are concerned with the dept of your cabinet. Check
this link for some pictures of a partly decased Dell monitor to see what I mean. I'd also recommend to look for the service manual of your monitor so you can get a more detailed view at it before taking it apart. This is how I learned that my monitor is fixed to its tilting base using a single screw.
With this knowledge, I plan to remove the tilting base and screw the monitor itself to the MDF shelve, which will be set at a proper angle.
Try to see if your monitor uses a similar stand. I think it could simplify things quite a bit.