@Nongnuch *******
I feel this part kinda answered what to do NOW.
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Wherever you are staying will need to file the TM30 for you.
Everytime you sleep somewhere else, you are supposed to get a TM30 filed.
Rented accommodation is fine for visa applications.
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But I guess this might have been a bit too unspecific not mentioning that the landlord of the rented home needs to supply documents.
The other part touched on the last question in OPs post.