If you're interested in studying deep philosophical issues, you might want to consider studying philosophy.
I can tell you that there is no room in basic neuroscience for deep philosophical issues, and as for cognitive psychology, I suspect that they spend most of their time thinking about psychology.
I can tell you there is a BIG room in basic neuroscience for deep philosophical issues. E.g. studying the neural correlates of consciousness (Logothetis . etc.) would definitely shed light (and already has) to the "deep philosophical issue of consciousness". A lot of mental functions are considered "deep" (e.g. thinking processes) but with cog. neuroscience, neuropsychology and psychology you can conduct some clever studies and see what's going on under these complicated processes . What you possibly mean is that there is no room in "applied neuroscience" for these "deep (-er) philosophical issues" which is true (but there still could be some philosophical implications)
Yes, of course cognitive psychology spends most of the time thinking about cognitive psychology (doh!) which means conducting experiments on perception, attention, consciousness, thinking, language, emotions and memory. The experimental ("scientific study" not philosophical) of these processes is called "cognitive psychology". Sometimes, when computational models are developed in order to explain the experimental findings it is called "cognitive science". When you conduct these experiments on brain damaged patients (because you want to see what mental function declined because of the brain damage) it is called "cognitive neuropsychology" (when you apply some of this knowledge for the clinical-cognitive- assessment and cognitive rehabilitation of neurological patients it is called "clinical neuropsychology") . Lastly, when you conduct these experiments on (usually but not always) healthy volunteers while you have their brains scanned with fMRI, PET or by using cog. evoked potentials it is called "cognitive neuroscience" (which has boomed big time after the 90s).
"Systems/behavioural neuroscience" ( also known as "biological" or "physiological" psychology in the past) is also a related field, one which conducts similar experiments-but on animals (and usually employs lesion studies or single-electrode recordings). These fields sum-up a lot of modern basic neuroscience (and at the same time some fields of modern "basic psychology" since they are interchangeable with some fields of basic neuroscience at times). Add "molecular/cellular neuroscience" (which is not always directly related to cognition/behaviour/mind but to the detailed cellular/molecular intricacies of the nerve cell ,something which of course can have cognitive/behavioural implications e.g. "long-term potentiation" and its relation to long-term memory-formation, or various molecular mechanisms of synaptic plasticity, dendrite growth, nerve repair after damage etc.) and you get the whole picture (of modern basic neuroscience).
So, yes, science in general-and experimental psychology and neuroscience in particular- have a lot to do with the "deep philosophical issues" that the threadstarter talks about, even with the "wildest" ones such as consciousness, free will and religious experience. See the neuroscientist's Benjamin Libet's experiments on conscious free will for example
http://en.wikipedia.org/wiki/Benjamin_Libet
Or the experimental psychologist's Daniel Wegner experiments on the illusion of conscious volition
http://en.wikipedia.org/wiki/Daniel_Wegner
Or the neuropsychologist's michael persinger on the field of "neurotheology" (and what brain structures and functions are related to various forms of religious experience, how temporal-lobe epilepsy can generate mystical-like experiences etc.), or kosslyn's cognitive neuroscience experiments on visual mental imagery. There are countless examples which demonstrate how modern (psychological and neuro-) science tackles some of the "deepest philosophical issues". It is similar to how modern cosmology/theoretical physics dwell in "deep philosophical fields" as well (e.g. what is reality? was there a beginning? etc.).