I am interested in knowing what kind of fossils we would find if we were to drill horizontally through the mountain and what we would find if we were to drill vertically. Would we find anything interesting other than the fossils?
The summit of the mountain is made up of 400 million year old sedimentary rocks laid down long before the uplift 50 million years ago. These rocks contain marine fossils including Trilobites and Crinoids. If you drilled down from the summit you would then encounter older metamorphic rocks, the upper parts of which contain traces of fossils, and then in to gneisses and granites.
Looks like some basic hints are necessary (as a complement to @AndyM's answer):
stratigraphy usually goes from younger to older when going down.
there'll be little chance to find any macrofossils in rocks that formed before sufficiently complex life was around.
there'll be little chance to find fossils in rocks that underwent metamorphism, that is have been in pressure/temperature regimes that aren't conducive for their preservation, even if they initially were present in the pre existing rocks.
'trace fossils' are not small remains of fossils but fossilized traces ('footprints'). Very rare thing.
it may be possible to find fossils in overlaying, younger sediments that formed during or after the uplift or were trapped or transported in depressions, but that's not the point of the question I think.
So, below the uppermost formations around the summit you'll likely find nothing of interest in the sense of the question when drilling down.
Will provide sources on specificically focussed request, but this isn't top notch geoscience.
Mount Everest is a Mountain Made from the Indian Subcontinent Colliding into the Asian Continental Plates, You are most likely NOT to find fossils because there was not a mount Everest during the Dinosaurs but Maybe some fossils could have gotten lifted by the collision but would be torn apart and scattered. according to [Source], The Himilayas formed 40 and 50 Million years ago, Which was around the end of the cretaceous period, which ended some 65 million years ago.
You will find Material and metals brought up from the bedrock of the Continental Crust, The Himalayas would eventually start getting weathered which would expose some metals.