If the word in the blank starts with a vowel, the "pe" is pronounced as part of the first syllable; e.g., Welcome to America = Lí niuspe sushli p'America.
"Pe piki" is "at the time when..."
"Piki lun" is a noun phrase, not an adverbial phrase. It could be the subject or object of a sentence, but to get the meaning desired for this sentence, it needs to be the object of a preposition.
This might make sense, but since it's talking specifically about time, pe piki would probably be used instead.
When used with an object (rather than a location), pe means "in".
If you want to clarify or focus that the cat is inside the box, and not any other relation to it, you can add the word "tius".
With the preposition pe "in, at", a direction specifies that the item is in that direction, but not necessarily close by.
This is similar; it uses sapchem as an adjective with its comparative form. Unlike the pe form, this only talks about one axis; that is, for example, the cat may also be far above the box or far in front of it.
...this also specifies that it's far away. (TODO is this right?)
Multiple direction-type words can be used together.
The word for "between" acts sort of like a conjunction, going between the two nouns.
Ípwenk usually refers to time, but it also has a metaphorical extension into space meaning "beyond"/"past".
TODO figure out/describe the word tas.
TODO that is.
TODO figure out what conjunction should go where "but" is. Also "with a heavy cudgel."