But did plants and trees of 300 million years ago sink so deep?
300 million years ago (Carboniferous) was a time when large amount of organic carbon was buried to a depth of several kilometres and formed large beds of coal. The pressure and temperature was only moderate (which is why you have coal and not diamonds).
Diamonds form at depths of hundreds of kilometres, and the source of the carbon can be any carbon. Inorganic carbon that was just there in the mantle, or organic carbon that came from any time in the Earth's history, not necessarily from the Carboniferous. So how do you get organic material hundreds of kilometres deep? The answer is subduction. I recommend you view this short video that does a terrific job in explaining of some of this stuff works:
there could be other exo-planets in the galaxy that are made of
Not quite. Not "made of diamonds". Just planets that have a lot of carbon. In our Earth and the other rocky planets in our solar system there is a lot of oxygen. When you take oxygen and bond it with other elements (silicon, magnesium, iron, aluminium, etc) you end up having rocks. What would happen if a planet would have more carbon than oxygen? In that case, it would be carbon bonding to the elements, not oxygen. You would have rocks made out of carbides, instead of rocks composed of oxides. You probably know carbides: these are usually very strong materials and they are used in industry whenever you need to do hard mechanical stuff.
Anyway, what would happen if you had more carbon than needed to bond with everything else? In that case you would have free carbon that in certain depths in that specific planet would be just in the right condition to form diamond. But this diamond will most likely be mixed together with all of the other carbides so I'm not sure if you will have huge blocks of pure diamond. But who knows?
Some more reading in this article.