The answer to your stated question ("Does anyone study non-paracompact manifolds?") is certainly yes. Here are a few papers which do just this:
Gauld, David.
Manifolds at and beyond the limit of metrisability. (English summary) Proceedings of the Kirbyfest (Berkeley, CA, 1998), 125--133 (electronic),
Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999.
http://www.emis.de/journals/GT/ftp/main/m2/m2-7.pdf
Among other things, Gauld references that there are two paracompact and two nonparacompact 1-manifolds, and $aleph_0$ paracompact and $2^{aleph_1}$ non-paracompact 2-manifolds. (That's a lot!)
Foliations and foliated vector bundles, 1969 MIT lecture notes
http://www.foliations.org/surveys/FoliationLectNotes_Milnor.pdf
Milnor entertains non-paracompact manifolds. In particular he constructs a (necessarily non-paracompact) surface with uncountable fundamental group. Milnor also says: "The main object of this exercise is to imbue the reader with suitable respect for non-paracompact manifolds."
Balogh, Zoltan; Gruenhage, Gary.
Two more perfectly normal non-metrizable manifolds. (English summary)
Topology Appl. 151 (2005), no. 1-3, 260--272.
The existence of perfectly normal, non-metrizable (hence non-paracompact) manifolds is shown to depend upon one's set-theoretic assumptions.
And so forth. I could find 10 more papers without much effort. I'm not sure I could find 100. (A MathSciNet search with "manifold" and ("nonmetrizable" or "non-paracompact") in the anywhere fields doesn't return many hits.) So some serious mathematicians take non-paracompact manifolds seriously enough to write some papers about them. On the other hand, although one could use more complimentary language than "extremely eccentric", your lecturer's take on non-paracompact manifolds seems to be an accurate reflection of how most geometric topologists feel: they seem mostly to be used as a source of counterexamples and to be of interest to general and set-theoretic topologists.
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