In terms of width of Y axis plate then 200mm is reasonable to reduce racking of the Y/Z axis assembly as this spreads the Y bearings apart a bit. Some people make the plate have 'ears' where the Y bearings mount which can help with clearance and assembly. Not mandatory as it depends on the other details of the design, but worth a look.

I went with less than 200mm spacing to get more working space as it is primarily a wood cutting machine. There is no problem with mounting a 150mm wide plate to this to hold the spindle. You can use the real estate at the sides to mount DTIs, laser pointer, dust shoe etc.

In terms of height of the Y axis plate only needs to be as high as the Z ballscrew, fixed and floating bearing, and stepper connection dictates. You need to draw these to see how tall it has to be to give the required travel on the Z ballscrew nut. In practise this could be close to 240mm but there is no specific requirement to match the moving Z axis plate in height.

Bearing sizes - I used 15mm on the Z (but wish I'd used 20mm as they are much stiffer), 20mm on the Y (which I'm happy with) and 16mm SUPPORTED rail on the X (wood cutting machine). If I was to use profile Hiwin style rail then I would use 20mm as a minimum.

The way you have mounted the gantry to the X bearings looks poor, as seen in the front view picture. A very long cantilever plate which is OK for wood, but not ideal for aluminium. You should redesign this, or add a further connection up to the gantry, or raise the X axis to meet the gantry directly.