Otan sateenvarjon mukaan, etten kastu matkalla pysäkille.

Because the umbrella is the object and here it’s treated as a total object (something you take as a whole). In Finnish, that often shows up as -n (genitive/accusative-like form) in the singular:

• sateenvarjo (basic form) → sateenvarjon (object form here)

Sateenvarjoa is partitive, used when the action is incomplete/ongoing, or the amount is “some/indefinite,” or in negatives. For example:

• En ota sateenvarjoa mukaan. = I’m not taking an umbrella with me.
• Otan sateenvarjoa esiin. (context-dependent, could imply “I’m getting at the umbrella / handling it,” not taking it as a complete “item to bring along”)

mukaan means along (with me), with me, to take along. It commonly goes with a noun in the genitive (-n):

• sateenvarjon mukaan = (take) the umbrella along
  It’s a set pattern: [something]-n mukaan.

etten is a fused form meaning so that I don’t / in order that I don’t. It comes from että + the negative verb en:

• että + en → etten
  It introduces a subordinate clause expressing purpose (especially in the negative).

Because in Finnish negative clauses use a special pattern: the negative auxiliary carries the person (en, et, ei, etc.), and the main verb appears in the connegative form (no personal ending).
So:

• positive: kastun = I get wet
• negative: en kastu = I don’t get wet
  Inside this sentence, etten kastu = so that I don’t get wet.

Yes, but the meaning and grammar shift slightly:

• etten explicitly matches I (1st person): so that I don’t…
• ettei is more general: so that (someone) doesn’t… / so that it won’t…
  In many contexts, people still use ettei even when the subject is “I,” but etten is the clearest match here.

Yes. These are common alternatives for purpose:

• jotta en kastu = so that I don’t get wet (very common, transparent)
• jotten kastu = a contracted form like etten, built from jotta + en
  All are understandable; etten kastu is a neat compact option.

Because Finnish typically separates the main clause and a subordinate clause with a comma. Here:

• Main clause: Otan sateenvarjon mukaan
• Purpose clause: etten kastu matkalla pysäkille

matkalla is the adessive case (-lla/-llä). With matka it commonly means on the way / during the journey:

• matkalla = on the way, en route
  So etten kastu matkalla pysäkille = so that I don’t get wet on the way to the stop.

Because pysäkille is allative (-lle), meaning to/toward the stop (destination).

• pysäkille = to the stop
• pysäkillä (adessive) = at the stop
  So this sentence focuses on not getting wet while going to the stop, not while waiting at it.