Do the Swedish/Danish authorities issue Schengen tourist visas at the Oresund Bridge?

I think you’re misunderstanding a key point about Schengen which is free travelling across borders — to the extent that the only thing telling you you are crossing a border is a sign saying ‘Welcome to Schengen country X.’ Unless, of course, you’re travelling by train in which case there is no such sign. You either realise you crossed the border by the language the conductors are using first changing, by a notable landmark (river, sea, pass) or by noticing that the railway track looks different (e.g. Danish railways generally having rusty catenary masts).

Therefore, when taking a train from Sweden to Denmark or vice-versa, you only notice you’re crossing a border because you’re ‘flying’ across the sea. At normal times there are no passport controls, no immigration booths, no nothing. You simply get on a train at Kastrup (or at any previous stop in Copenhagen) and get off again in Hyllie (or any subsequent stop in Sweden). This is no different from a train journey Copenhagen–Odense.

Now with the large numbers of migrants in late 2015, border controls have been temporarily reintroduced. (I am fully aware that ‘temporarily’ sounds weird considering it is almost 2017 already.) However, these checks are as follows:

• At the final station in Denmark, staff from the train companies quickly verify your documents that you have a valid visa or are otherwise eligible to enter Sweden. If you don’t, they don’t let you board for not complying to their terms of carriage.

• At the first stop in Sweden (Hyllie), Swedish immigration police boards the train and again checks that everybody has a valid document. No visas are issued, nothing is stamped.

Therefore, there is no way of acquiring such a stamp or anything by entering Sweden from Denmark by train. In fact, you may even get into trouble there if they see you overstayed your visa — that would result in quite the opposite effect as far as I would guess.

If you want to get such a stamp/visa (if they are even issued to you on arrival, which I strongly disbelieve) by flying into Denmark, you will have to make sure that you actually came from outside the Schengen area (e.g. the UK, the Balkan, Russia, overseas) otherwise again you will not walk into any immigration control.

What you usually get when crossing a Schengen external border as a New Zealand citizen is not a visa but an entry stamp. It's important to note the difference to understand your situation. Schengen visas can only be issued at the border under a very limited set of circumstances (e.g. family members of EU citizens who could not apply for one in advance). But as a New Zealand citizen you do not need and cannot get a Schengen visa.

During a regular visit, you can simply enter the area without a visa for up to 90 days and would get a stamp documenting the day of entry. Your situation is a little more complicated because you are currently staying on a residence permit and the 90-day visa-free stay would only start at the end of this permit. That's completely fine in most cases but that means you won't have a stamp with that date on it and some countries do apparently insist on people getting such a stamp. The only way to do that is to leave and reenter the Schengen area, for example through a short trip to the UK.

I don't know that getting a stamp would really be necessary to stay after the end of your residence permit but to the extent that it would be, then merely staying in Germany or entering Denmark could theoretically be problematic as well because you would also do that under the very same visa-free visitor status. Furthermore, if you really do need a visa to marry, then a Schengen visa is not enough (and, again, you cannot get one).

In practice, all this means that:

• You might be just fine as it is, check with the Danish authorities before doing anything costly.
• If you need an entry stamp to start a period of visa-free stay, you need to go to the UK, Croatia, Turkey, etc. before the expiry of your residence permit. There is no scenario under which you could be allowed to remain in Germany and visit Denmark without implicitly using that status already so, in principle, your intention to marry does not really make a difference. And you cannot get such stamps at the border with Sweden.
• If you really need a visa, it would mean that the usual visa-free visitor status is not enough to marry and a round trip to the UK would not help. What you would then need is something else than a Schengen uniform visa (some Danish marriage visa?) and almost certainly cannot be obtained at the border, whether the Swedish border or a proper external border like the airport in Copenhagen.