🇩🇪 German elections (federal & EU level)

[Coffein00]

Why didn't they go for Red-Red-Green in 2013?

The short answer:
The complete leadership ruled it out before the election back then repeatedly. The reasons they were listing were, that the Linke is unreliable and incompatible with the foreign policy of the SPD. Also a minority government tolerated by the Linke was ruled out.

The longer answer:
As we all know, these things do not need to hold after elections and the Linke actually tried to convince the SPD to change their mind.
You have to consider though, that the majority of a Red-Red-Green coalition would have been 5 seats (RRG having 320 and CDU/CSU 311). This is usually not that big of an issue, if the partners are really trusting each other and have similar policies. For example in NRW there is a Black-Yellow (CDU+FDP) coalition governing on state-level with a majority of just one seat and it works.
But in the case of RRG, there would for sure have been defectors. On the SPD side a few, because they did not want to have a coalition with or tolerated by the Left on principle. And in the Left/Linke there were still some SPD-hating MPs, which are mostly coming from Western Germany and were members of the WASG (a party which splintered away from the SPD during the Schröder-government and then united with the PDS to Linkspartei.PDS, which was later renamed to "Die Linke".) They were still angry at the SPD and there would have been a huge risk that some of them deflect, too, just to "show it to the SPD". (If you are interested in this topic google "Andrea Ypsilanti", to see an example).
Maybe it could have worked, maybe it would have failed. I say the chances that a Red-Red-Green coalition would have governed for the full term are pretty low, but still around 10%-30% (If I consider that they would have had to deal with the refugee-wave in 2015, these numbers probably are to optimistic).
Furthermore the SPD feared that they would be seen as breaking their word or being just hungry for power for every price. Plus you do need to consider how favorable Angela Merkel as chancellor was at this time.
Either was, they all three would have dropped in the polls if they had formed a government together. With good policy over the 4 years and a low amount of scandals, they may have recovered.
But again, there are a lot IFs involved in this.
Also the SPD did get the Union (CDU+CSU) to concede on suprisingly many of the SPD core issues, like the minimum wage, for example. So in the beginning the SPD thought they got a good deal there actually.

So from a strategic viewpoint the SPD certainly believed back then, that a GroKo would be the better choice for them.

[Coffein00]

And another topic I want to weight in is the proposal for the SPD to go the Danish SocDems way and adopt much harder immigration policies and on the same time go the Corbyn way to win back voters from the Left party.

We need to keep in mind, that (in my polling average/trend) the SPD is at 12.2%, the AfD at 12.9% and the Linke at 7.6%. This sums up to 32.7%. Even if they could manage to get a 100% of both AfD and Linke votes and on the meantime not lose a single voter, because of the harder immigration policy to the Greens or maybe FDP, in case these voters are against the more left policies, for example people from the Seeheimer Kreis(neoliberal and more conservative wing of the party). So lets assume this for one second, even though it is totally unrealistic. They still would not have enough support to form a government. They would need another 14-16%.

The Greens would not go in a coalition with this SPD. They were the most consistent party on their immigration policies during the 2015 refugee-wave and they are clearly against significantly hardening these policies. This is also a factor which helptem to their current strength.

The CDU won't cooperate with them, because of their new further to the left social-and economic-policies.

The FDP would very likely not have enough votes to help them get a majority. And besides that, they would not want a coalition for the reasons the Union won't want one AND the reasons the Greens won't want one.

The result would be Green-Black / Black-Green. So basically nothing won, but risked a lot. In Denmark the situation is different, as many people in this thread already pointed out. But here in Germany, the Greens would not think twice if they should stand with their "red bloc" partner SPD even though their immigration policies are completely contrary or if they team up with the CDU (in case they did not move as far to the right on these issues, too). And if necessary even the FDP would likely join a Jamaica coalition this time. They certainly learned their lesson from 2017.

I do not want to give the SPD any suggestions and I'm glad I do not have to decide which direction the party should take.
My first guess (and I probably have to say I'm not a SPD voter or member) was that it might help them in a long turn to nominate Kevin Kühnert. This way they would keep open the door for an Green+SPD cooperation in the future and they would have a somewhat likeable (compared to people like Olaf Scholz, Hubertus Heil, the three temporal leaders, Oppermann, Gabriel etc.). He could also inflame a fire which would help the whole party getting more popular again. The left-politics of Kühnert would be weakened anyway, but probably still would have the potential to excite people who vote Left, "Die PARTEI", Green and most important non-voters to vote for the SPD. Yes, they might lose voters to CDU and FDP, but therefore they would have the option to build a coalition with the Greens after the election.
When I talked about this option with a friend on the morning Nahles stepped down, we both were having the same idea and both basically said "what do they have to lose".
But again, I'm glad I'm not between those who have to decide about this...