Are bounty reputation points, being assigned to unanswered question, returned back to the issuer of the bounty? If "yes" - when? If not - why? Alternatively (instead of direct "refunding") in the case, when there are no answers at all, the issuer's may be should be given the option of counting this "spent" points towards doubling the bounty or towards transfer of that bounty to another question? Another option could have been extending the time on the bounty?
No, points placed as bounty on a question are not refunded in the event that the question receives no answer.
On the reason let me quote myself from another discussion about the merits of refunding bounties.
The reason that a bounty costs points is two-fold. On the one hand the points are a direct incentive for somebody to answer. Yet, on the other hand, the bounty gives significant additional visibility to the question.
The second point seems at least as important as the first one. It is this additional visibility for which you have to "pay" no matter what.
Bounties need to "cost" something to stay effective. If they would be free, somebody could put a bounty on essentially unanswerable questions all the time, under-cutting the effectiveness of bounties.
On your specific proposals:
[W]hen there are no answers at all, the issuers maybe should be given the option of counting this "spent" points towards doubling the bounty[...]
This proposal seems like a reasonable one to me. A point to consider though is that "no answers at all" is a criterion that can cause problems (e.g., askers complaining more about not satisfactory answers because it makes them lose the "no answer" option). It may also not be needed. It will not be that common that something is answered to an asker's satisfaction and they still want to prolong the bounty.
or towards transfer of that bounty to another question?
By contrast I do not like this one, as it will inflate the number of bounties.
Another option could have been extending the time on the bounty?
Maybe, but I prefer the doubling idea.