The first solution Duo gives "Do you've a dollar...." is unnatural. Here is an example of the wrong usage og "...'ve..."
This happens automatically when we have "you" and "have" next to each other in the incubator, unfortunately nothing we can do about it right now, not sure why it's suggesting it as a translation though
have /have got are the same.
It depends on where you put the "got". What was your English translation?
If you wrote, "Have you got a dollar that I can have", that would be the way many English speakers would say, and should be accepted.
Have you a dollar that I can have? Should be allowed!
is it a common way to ask for money? or they use something like "kan jeg låne..."?
Normally you would say kan jeg låne, but in the sentence above you don't have to give it back, and as such it's not "loaning the dollar".
The (female) TTS pronounces 'jeg kan' roughly like 'jejke', that's not correct, is it?
She's actually saying - jeg ka' - they often drop the last consonant.
Ok, but my focus was mainly on the pronounciation of 'jeg', which sounds like /jai/ here, when I thought it should be /je/
Have you a dollar I can have is much more everyday, and means exactly the same thing as if you added a 'that'
If you swap in "Do you have"/"Have you got" for "Have you", I'll agree
Have you a dollar.... Should be accepted