Are k-spaces named for Kelley?

Engelking cites this paper as the place where $k$-spaces were introduced, though the author, David Gale, says the notion was first defined by Hurewicz. The $k$ probably refers to the German `kompakt'.

Answer to 2). No. Hausdorff means that the diagonal is closed in $X\times X$. The "correct" separation property usually added to $k$-spaces is "weak Hausdorff", which on $k$-spaces means that the diagonal is closed in the $k$-ification of the usual cartesian product.

Just to add a bit to the history, the first time the exponential law was given for Hausdorff k-spaces was I believe in my DPhil thesis, submitted 1961, see PtA available here, which was circulated to the obvious places. In my first paper, (1963), also available here, I wrote:

"It may be that the category of Hausdorff k-spaces is adequate and convenient for all purposes of topology."

In my second paper, (1964), I used the category of Hausdorff spaces and functions continuous on compact subsets, and showed it was what we now called cartesian closed. (My thesis contains an attempt at showing the idea of what we now call monoidal closed, since in my thesis I had lots of internal homs and associated products, usually a tensor.)

I did not understand final topologies at the time and so did not come up with the definition for the non Hausdorff case, but you can find that in my book Topology and Groupoids.

Several people wrote about the non Hausdorff case, but an important application is to fibred exponential laws which were developed by Peter Booth following some ideas sketched by R. Thom. See Booth, Peter I. The section problem and the lifting problem. Math. Z. 121 (1971), 273–287.

I also wonder whether the category defined in

Johnstone, P. T. On a topological topos. Proc. London Math. Soc. (3) 38 (1979), no. 2, 237–271.

is indeed adequate and convenient for all purposes of topology, in particular can cope well with fibred exponential laws, since being a topos is a stronger condition.

There are also the purposes of analysis, for which see

Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis. Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997.

So the term "convenient" has had a good run!

See also convenient.