Is quantity a dimension? We believe that time is a dimension and that $x$,$ y$, $z$ are dimensions in space. Is quantity a dimension like these? And if not, how do we have dimensionless numbers (like $e$, $\pi$ etc.)? 
 A: quantities have a dimension, and units, exept when they don't.  A definite path has a length, so it has the dimensions of length, and you can pick the units to be centimeters, and get the length to be a number of centimeters.
$\pi$ is dimensionless because it is a ratio of two lengths, and the units cancel out.  for this reason, it is the same whether you used centimeters or inches.  It is a pure, dimensionless, number.  So are sines and cosines.
Quantity is not quite dimensionless, but it is not one of THE dimensions that make up space and time (this represents an ambiguity in the meaning of the word "dimension").  Quantity has units, so it is not a pure number.  The number of neutrons is measured in units of neutrons.  If you had twenty neutrons, it would be 20 (a pure number without dimension or units) * neutrons (a unit with a dimension).  If you changed units to measure that dimension, say, quarks, it would be 60 quarks, where now quark is a unit measureing one-third of an neutron---  note, and this is deliberate, this meaning of quark is different than if you were trying to measure quarks.  Then your dimension would be quarks and your unit would be quark, and it could refer to any kind of quark indifferently, they wouldn't have to group together correctly to always amount to an integral number of neutrons.  Or you could change units to "one-half of an neutron", which doesn't even have any separate physical significance, and then although the dimension is still "neutrons", the change in units leads to your measuring twenty neutrons in terms of 40 "ha'neutrons" (kind of like a ha'penny).  
A: Pi and e are quantities, proportions really. But because we multiply/divide by identical measurements; ie: length for pi; the measurements cancel out and we're left with a pure number, dimensionless as it is. X, Y, and Z are representative of unspecified quantities, so i think that's covered..
A: Since you mention time and x,y,z as dimensions in space, i think you mean the geometrical dimensions and not the unit dimensions.
So i answer this question with the above in mind.
Yes quantity (with or without units) can be a dimension in some space.
Since space is sth related to a definition (and the associated dimensions of this space).
One can have for example a dimension of color or a dimension of  number of particles (a unitless quantity), in some representation of a physical system.
Indeed in Hamiltonian mechanics the space of the physical system under study (phase-space) can have many dimensions (e.g 12).
Also popular theories beyond the standard model (eg String Theory) do in fact assume 11 dimensions.
What would be the physical (or maybe realist) meaning assigned to these dimensions is another matter.
