Imagine how silly this sentence would have been 30 years ago...
Does this mean something like "He was texting?"
Yes, it can mean both that he is texting, but also that he is literally writing something on his phone with a pen.
But it's most likely to be understood as texting
Is there another way of referring to texting?
I can only think of the unformal way: "Han sms'er" (He smss)
It accepted "texting" for me
'the boy wrote on his telephone' got corrected to 'the boy was texting on his telephone' - should my answer be correct?
This lesson here is about the Danish progressive construction, which you can see at the use of "at gå og..." So your sentence should use the English progressive form, too. "The boy was writing on his telephone."
Is there a firm rule whe you should use "sidder og" etc.? Or is it a matter of memorization
No firm rule. Just pick what's currently happening (he's walking while texting in this case) or what makes general sense (so not something like "hun sidder og svømmer").
Challenge of sitting and swimming simultaneously accepted. Now attempting to breast stroke while in cannonball shape. :p