Is there a side that is used primarily for melody or doesn't it matter?

There are two common styles of playing - one is to play melody mostly on the right and use the left for accompaniment with chords and bass (albeit not vey low) notes. That's usual for morris tunes. The other is to use both sides and play just melody. You'll get some lower notes that way, and that's the usual style for Irish music.

2. Can it be cross harped like a Harmonica?
Yes, as you've observed it's only a harmonica with its own lungs. The things you can't do so well as a harmonica are bend notes and get all those chages of timbre by making resonant shapes with your hand.

Is there a specific technique of playing to accompany singing?
Commonly the morris style - play tunes and chords, but with practice you can make some very refined versions of that. I often use it in a contrapuntal style with disctinct internal melody lines making up the harmony, but that's not easy - when I do that I'm not singing at the same time but accompanying someone else!

Really there are no rules - those are just some examples of what's commonly done.

Can anyone suggest a few simple tunes to start with?

You found morris tunes - they are a good start!
A book of simple tunes for any instrument is a good start. A non -obvious but good example is Julian Goodacre's books of tunes for pipes - all with necessarily limtied range, no accidentals and not too flashy.
Dave Mallinson has lots of good stuff too. Personally I'd recommend any of the "English Choice" books but note there are also copncertina specific tunes book and tutors if you want them.

Some of them may not work so well on the concertina, and some will need more than a 20 key instrument, but there are plenty that will work.

Anahata