Hmm, organic chemistry is most certainly not my forte (then again, what is?), but I had a re-read of my lecture notes and hopefully can provide some explanation:
I find that often with these 'choose your reaction mechanism' type questions, in general there is no 'proof' of your answer being right, unless you actually carry out the reaction with precise control over experimental conditions, and see what you get (after all, that's how science is!).
So, what we do here is look at general principles in order to make the best educated prediction we can. All this being said, I agree with your conclusion that an E2 mechanism is most likely. You are right that Sn2 is a very unlikely reaction mechanism, due chiefly to the bulky groups on the substrate, and also due to the bulk on the potential nucleophile! Why? Because in an Sn2 reaction mechanism, the nucleophile must attack along the same 'line' as the C-(leaving group) bond, but from the other side (we call this 'backside attack'). The transition state actually has the attacked carbon with 5 things attached to it (in a trigonal bipyramidal fashion) - the nucleophile and the leaving group form the 'axial' positions while the three other original substituents (in this case, methyl groups) form the equatorial positions. Then, the leaving group leaves, and a chiral inversion occurs. In order for this to work though, the nucleophile must be able to position itself for the backside attack, and that is very difficult if the attacked carbon has lots of bulky groups attached to it, and even more difficult if the nucleophile is itself bulky! (Unless you were enquiring about whether to use the term 'base' or 'nucleophile', in which case, because you are talking about a substitution reaction, you would refer to the attacking molecule as the 'nucleophile').
You also raise a good point that the C-Cl bond is not so easy to break, which is an argument against both an Sn1 and E1 mechanism. I'd like to add an additional point here: notice that the potential nucleophile/base is quite bulky, which would tend to favour elimination over substitution. Furthermore, it is also quite a strong base (the conjugate acid would be a tertiary alcohol, which is a fairly weak acid), which also favours elimination over substitution. Therefore, I would claim that an elimination reaction is more favoured over a substitution reaction here.
Then, your point regarding the difficulty of the Cl- group spontaneously leaving can be used to predict an E2 over an E1 reaction. I would again offer the strength of the base as a supporting point to this - if we have a strong base, it really would like a proton, and why should it wait around for the Cl- group to leave, when it can abstract a proton from the neighbouring carbon? Finally, just as a quick check, is there an abstractable proton that is anti-periplanar to the leaving group? Yes there is - of the 3 hydrogens on the neighbouring carbon, one is anti-periplanar. So therefore, an E2 mechanism is both plausible and likely!
Turning to the suggested answers, we see that option G results from a substitution reaction. So that's unlikely. I'm a bit confused by your statement, " E will be formed once the base deprotonates one of the beta-carbons". An E2 reaction is a concerted process - which means that basically everything happens in one relatively smooth motion, and you end up with your products. So to me, this means essentially that by the time the base has fully abstracted the proton, all of the other stuff has already finished too. So, we should end up with the ketone (option F), plus a tert-butyl alcohol. The tert-butyl alcohol is not option E (note that the carbon attached to the O atom would need to be a tertiary carbon, and option E has a primary carbon instead). So to me, the answer should be F.
Hopefully this helps, and hopefully I haven't stuffed this up completely!