There's a particular style of joke/pun in Japanese that I'm not sure how to describe, so let me provide some examples:
(said by someone who has recently become a 神)
(said by someone who is looking for people to help out a basketball team)
(following a conversation about giving mochi to classmates and trying to come up with a catchphrase)
Now that we've established what I'm talking about, my questions are as follows:
Is there a name for these jokes? (Would you just call them ダジャレ?)
What is going on grammatically in these jokes? This doesn't seem to be one of the simple/compositional usages of だけ＋に found e.g. here at dictionary.goo.ne.jp.
Is there a rough English equivalent of these jokes? I've seen translations that basically introduce a pun related to the topic of the joke and then append "No pun intended", presumably corresponding to the だけに part. Of course, joke translation is a peril-fraught thing, so I'm not placing too much stock in this.