Here's the deal on the local trains.
You can take your bike on any local train at any time but you need to apply at least one hour in advance, In practice, I have always been let on even when I just rolled up a few minutes before but they may read you the riot act in some places Wanhua is really strict for some reason. Small local stations are a breeze. They want to check if the quota of bikes is already on and if the train is too crowded.
You also need to bag your bike. What constitutes bagging has been a bit controversial. I have resorted to big plastic bags for some time but a month or so ago coming back from Miaoli, I was sternly told by a very pretty conductress that it had to be a 'proper' bike bag. A few weeks later I read in the paper that the TRA officials say that plastic bags are fine as long as the bike is actually in them.
I think this sums up the deal. The power that be want bikes on the trains. The local conductors who have to deal with crowded trains and annoyed commuters don't like it, will try to adhere to the rules, and may impose additional ones.
Just ship you bike for longer rides. It's a lot less hassle in the end.