When you don't have a Danish keyboard, the 2nd best thing to do is to write the 3 special letters we have in Danish like this:
æ = ae
ø = oe
å = aa
No worries, though. Many foreigners misspell the names of towns, castles, etc. - including many varieties of æ, ø and å. But, since some of the names are written very similarly, we just have to ask to be sure we're talking about the same thing (for example Fredensborg, Frederiksborg and Frederiksberg are 3 different places in the Copenhagen Region).
I think a ferry is needed for a real island experience. A lot of the time, the small island charm disappears when a bridge or tunnel is built (infrastructure like that is very handy, but has some consequences too). Denmark is an island nation, so ferries have been, and is still, an important part of the infrastructure. Many people use ferries every day, so taking a ferry in Denmark is an easy operation. You don't have to worry about that.
Another option would be Samsø. Take the ferry from Kalundborg to Ballen on Samsø, visit the island, and then take a ferry from Sælvig onwards to Hou on the mainland (or the other way around).
If you plan on using bicycles on Ærø (which is a great idea, by the way), I don't see the need for a car on your trip (you can use one, but it's not needed) - of course depending on what area outside Odense you want to visit, but the bus and rail networks can get you to most places on Funen.
I mean no offence on Odense, but I prefer Aarhus. For starters, it's bigger and has a lot of interesting stuff to do (like Aros, Den Gamle By and Moesgaard Museum). That is unless you're an H. C. Andersen fan, then Odense is an obvious choice.
If it was me, I wouldn't stay overnight in Helsingør, as this will require changing hotels, when it's not necessary for your itinerary. Instead, take the train to Helsingør and Humlebæk and then return to Copenhagen at night.
I think I wrote this in a very random order, my apologies. Hope it didn't make it too difficult to read :-)