Upgrade from v10 to v13 should be straightforward and it shouldn't cause any issues as you described if done correctly, as the database structure is very similar, if not the same.

I'd recommend you follow the Umbraco documentation and these handy videos and do you upgrade again. From your error, I believe it could be due to some missing Media Types, so you should check them (manually creating them might help resolve some of your issues but again, migrations should be done as described in the following videos.)

Handy resources for upgrades:

https://www.youtube.com/watch?v=aAgmq5Azn4s

https://www.youtube.com/watch?v=075H_ekJBKI

https://www.youtube.com/watch?v=_rQs0oO4uLA