You are allowed to bring your own luggage on the Swedish trains (you carry it on yourself and place it at either end of the carriage or on a shelf overhead) so if you pack your bike in a box or flight case there shouldn´t be a problem. And if someone asks what´s in your large suitcase, it is of course a piano or all your winter clothing...
Regarding starting times: You will need to use lights and reflectors (not sure if this this is the correct word in english, called "reflexer" in swedish), white forward, red backward, and yellow to the sides, if you start before 2:50 am. It used to be that a "late start" automatically meant no lights needed but as of last year,I think, "late start" means after 2:30. (Just don´t want you to be unaware of this as there is a lights check at the start)
Also: There are actually a couple of chartered trains between Stockholm-Motala and Göteborg (Gothenburg)-Motala where you live onboard during the stay in Motala. They are run by two separate cycling clubs and usually fill up quickly but they might be a convenient sollution. Stockholm-Motala run by Fredrikshof cykelklubb
Hem - Fredrikshof Gothenburg-Motala run by Lygnens Vänner
Lygnens Venner - Den glada cykelklubben