I thought it was twice in a row. I thought I had read it as "win the title and defend it next year".
It is mix-and-match. Any two of the specified titles, even if different, in any order and at any time, not necessarily consecutive (though I vaguely a debate when the system was first introduced as to whether titles earned prior to the new system could count - I can't remember what was decided, but it's moot now).
And according to the main source, NK's website, it's "各2期" for the previous 4 titles. My understanding is that, for "twice any time", it should be "2回", (回=counter for times)
I can't remember the figure now but I recall seeing lists of counters or measure words in Japanese and Chinese that claim at least (?) 300 and 600 respectively. But certainly a lot. Not all are current and they seem to be (very) slowly dying out.
期 is correct for things that have known and usually regular stages/terms/edition/phases (e.g. illnesses, sessions of parliament and lots of other things).
We have quite a few measure words in English you may not be aware of: head of cattle, brace of pheasants, pounds, pints, shots, and the pecks, bushels, gills of my childhood. And a host of collective counters. Anything for sensible people to rely too much on numbers. We had a well-known one the BBC web site in the last day or so: a photo showing a "murmuration of starlings." Inventing collective nouns is a popular English pastime. What would the collective noun be for L19 contributors: an invisibility of contributors; a black hole of contributors; a silence of contributors; a figment of contributors? Over to you
