If we get married in Sweden, and don't register the marriage in Japan. What will happen?Nothing, as long as you stay in Sweden and just visit Japan once in a while.
But officially Japanese nationals are required by law to report important changes in their status (such as marriage, divorce, childbirth, etc.) within a few months (I forgot exactly how many) to the Japanese authorities, either to the city hall in Japan or through Embassy of Japan in Sweden (in her case).
Your Swedish marriage can be reported “after-the-fact” with the Japanese authorities (you don’t have to get married again according to the Japanese law). What I mean by “after-the-fact” is that my marriage in the UK is mentioned in my family register as:
“Ms. AK got married to Mr. So-and-so from (country name) according to the UK law on (date).”
Only in case the two of you ever, ever decide to come live in Japan as a married couple, it’s just easier if the marriage has been reported to the Japanese authorities too (but she can report later anyways). Also if you have a child and if you want the baby to have Japanese nationality…
By the way, if she reports late, there will be a fine of something like 2,000 yen (no criminal record, no worries – I did this myself – myself Japanese, my husband continental European, we got married in the UK where we used to live, did not report the marriage to the Japanese city hall until a few years later, when we decided to move back to Japan.)
Also what do you need to do to switch to another citizenship from Japanese, more than applying for citizenship in the new country?I don’t know the procedure for her to obtain Swedish nationality, but given that happens, officially she has to “renounce” her Japanese nationality, as Japan does not allow dual nationalities.
Q13 on this page:
http://www.moj.go.jp/EN/MINJI/minji78.html#a13She might want to check with the Embassy of Japan in Sweden.