Do electrons interact with gluons? I know the straightforward answer to this question is no: electrons are leptons which by definition don't interact via the strong force, gluons are the mediators of the strong force and hence electrons don't interact with gluons. 
That said, in this power point by D.Stump the parton distribution function is defined as (pg45, paraphrased):

$f_i(x,Q^2)dx$ is the mean number of the $i$th type of patron with longitudinal momentum fraction from $x$ to $x+dx$ appropriate to a scattering experiment with momentum transfer $Q$.

My concern is that $f_0(x,Q^2)$ is in general non-zero, with the subscript $0$ here referring to gluon. From my interpretation of the phrase appropriate to a "scattering experiment with momentum transfer $Q$" I would interpret this to mean that the scattering particle (an electron) interacts with these quarks. Is this interpretation correct - if so does this mean that electrons interact with electrons and if not how should this phrase be interpreted. 
 A: Electrons not only can interact with gluons but this interaction has been a routine measurement at electron-proton colliders. This happens through the following diagrams (at the lowest order in the strong coupling constant). Note that those are tree level (i.e. no loop). This is the leading-order contribution to the electro-production of heavy flavours, especially studied was the case where $q$ is the top quark.

A: One often hears about Parton distribution functions (PDFs) in the context of inclusive Deep Inelastic Scattering (DIS) analyses because, indeed, this was the first process to reveal the substructure of the proton.  
The interaction is the scattering of a electron off a hadron at high momentum transfer $Q$. A clean probe of the proton is an electromagnetic one, namely the photon, and it is this that carries the high momentum transfer. Usually the transfer is taken to be at a large scale not comparable to $\Lambda_{\text{QCD}}$ or the mass of the hadron so that the picture basically boils down to the probe interacting with a single partonic constituent of the proton. This then defines the parton model and an understanding of any kind of proton tomography is accessible through studying these objects called PDFs.  
These PDFs have probabilistic interpretations of finding partons having longitudinal momentum fractions $x$ within a certain range when probed at a particular $Q^2$. At larger and larger $Q^2$ the parton sea soup within the proton becomes more and more resolvable. 
In this iconic DIS analysis, at the leading order (LO), yes, the photon will only interact with the quark/antiquark constiuents of the proton but then the gluon contribution can come in at higher orders via an appropriate loop insertion for example. 
In some other processes amenable to description by perturbative QCD due to the presence of a large scale,  it is instead the gluon contribution that is at LO.
