I don't think this should be translated into 'nightly', because in English the adverb 'nightly' has the very specific meaning of EVERY night. If you say "The dogs and cats play nightly" in English it's taken to mean the dogs and cats play every night. The Esperanto 'nokte' does not have this meaning, it means 'by night' or 'at night'.
Oddly, this dictionary has "nightly" translated as "nokta" and "by night" or "at night" as "nokte". http://www.gutenberg.org/files/16967/16967-h/16967-h.htm#letterN
This dictionary has the same for "nokte", but has "nokta" mean "nocturnal". http://esperanto-panorama.net/vortaro/eoen.htm
....and this dictionary says that "nokte" means "dum nokto". http://reta-vortaro.de/revo/
Also, "on Friday" does not mean "every Friday" which would be "on Fridays" but means instead "sometime during the 24 hour day of Friday".
To Philip Davis below:
I think "every night" would be "ĉiu nokto".
nokte is an adverb and I have never seen a plural adverb.
I am learning though, just like you and I await someone more knowledgeable.
"Every night", as an adverb, can be "ĉiunokte". By the way, "ĉiutage" means "every day". See some examples here: http://bertilow.com/pmeg/vortfarado/principoj/frazetvortigo.html#i-o8c
About Phillip_Davis' question: an adverb can never take the plural ending -j.
Does this mean "The dogs (these ones) and cats (these ones) play at night" or "The dogs (these ones) and cats (in general) play at night"?
In english it's reasonable to assume that the dogs and cats are grouped together, with both being referred to by 'the', but is that also the case in Esperanto?
"Ili ludas nokton" would mean that they play "night" (which would presumably be the name of a game), whereas "ili ludas nokte" uses "night" as an adverb, meaning they play at night.
Similarly "ili ludas rapido" means they're playing speed (a card game), and "ili ludas rapide" is "they're playing speedily (or quickly)". Of course speed is, as the name implies, a game played quickly: oni ludas rapido rapide.
Good question. What do we call them in English? The politically correct term keeps changing. Here are some options.
The last two - especially the last one are kind of a joke - but it might be fun to use these terms.
That would have to be "dormi estas malfacile", because in Esperanto you can't have two ordinary verbs together. In English, the same word "sleep" can be a noun and a verb, so it is possible to say "Sleep is difficult". However, suppose the verb was "write". Just as you would never say in English, "write is difficult", so in Esperanto you never say, "Skribas estas malfacile". It has to be "to write" in English and "skribi" in Esperanto. Actually in the first part of your post, you had the right idea - you didn't put "jes, mia kato tre sxatas ludas nokte", but "jes, mia kato tre sxatas ludi nokte".