From reading your two posts, intuitively I feel like a Slav/Triangle combination will be quite a good fit for you. You can use the Classical Slav (possibly with the Chebanenko as an alternative to Avrukh's 3.Nf3 Nf6 4.Nc3 dxc4) as a main option when you want a solid, more strategic battle, and have the Triangle for when you seek a sharper or more original position. That way you can dodge the Exchange Slav if you want (with the 2...e6 3.Nc3/Nf3 c6 move order) and still get a very solid game with a Stonewall if White tries avoiding the complications with an early e3.

There are several good books about Nimzo.

For a quick repertour there is "Nimzo Indian move by move" by Emms.

Other usefull books are "Mastering the Nimzo Indian" by Kosten.
Sokolovs "Winning chess middelgames" also covers lots of Nimzo structures.

There is also a comming repertour book that combines Nimzo Indian with Bogo Indian but it has been delayed several times.

Thanks for all of your interesting and great replies. Some of the arguments here are totally spot-on. 

1. My absolute favourite player is Karpov and the books Karpov's Strategic Wins are just fantastic I find.
2. Why the caro-kann? What I like about the caro-kann? Really many Danes are influenced by Larsen's booklet "Solide Åbninger" from 1980 where the Bronstein/Larsen variation gxf6 was extremely popular at club level at least. Returning to chess a couple of years ago, I see that the above variation is now considered dubious. I participated in a couple of rapid tournaments and got crushed in the caro-kann from 2010 and onwards, because everybody played the Panov-attack or the advance variation which I encountered rarely as a youngster. Schandorff's book on the caro-kann from QC is definitely my favourite book on openings which I studied carefully a few years ago. Instead of the Larsen/Bronstein, the old Lf5 variation is nice with short castling on the black side. White gets a lot of space but typically the fields g4/e4 can be used by black and I like these positions which often start razorsharp but end in some unclear endings. Against the Panov, I enjoy the endgame line and find that black's chances are excellent. In the advance variations, white again gets a lot of space but here I like the grip of d5 which black often gets. In general, I guess that I like fixed pawn structures. Hanging pawns like in the Tartakower variation are not my strength but of course, if you play these variations, you will learn how to play them well, so this could be a way to develop as a player. As a youngster, I also loved the Najdorf variation in Sicilian but this just requires such a heavy knowledge of theory.
3. I completely agree on the point about the difference in the Pirc and the KID. In the Pirc, black really risks getting killed in the opening if he doesn't know his stuff. IMO, the same applies to the 4-pawn attack in the KID but in many cases each side has long pawn chains with chances on each side which is very interesting. As I see it, there is even quite a difference between the Modern and the Pirc where I clearly prefer the former because you, some times, can use the Gurgenidze variation against the Austrian attack but probably only as a surprise variation where you again typically have fixed pawn structure.
4. The Nimzo/QID could also be interesting to play and this was as already-mentioned part of Karpov's repertoire. However, for ages he seemed to be very happy to draw with black. I have the feeling that Karpov may have felt that playing chess with the white pieces was almost like serving in tennis where this is a huge advantage. Could you recommend a good book on the nimzo? I can see that chess.com has tons of videos on the nimzo. I think that I will watch a couple of them to see if this opening is something to my taste.
5. The slav. Must try this opening at least just a few times. A rapid tournament could be the perfect place to see if this could become a long-life passion.
6. The French/Stonewall could also be great to try. The Stonewall seems relatively easy to play with black without knowing tons of theory. I have just seen a couple of videos on the Stonewall and the opening seems great.

Again, thanks a million for your great comments 

Personally I've never bought into the similarities between the Caro-Kann and Slav apart from the pawns being on c6 and d5. I think it's the same with the King's Indian and Pirc, same pawn structure but the ideas and feeling when playing it are quite different. 

I think the Caro-Kann has a more positional and manoeuvring feel whereas the Slav is more tactical and static. Openings that have a similar feel to the Caro-Kann imo would be the Nimzo partnered with either the QID or QGD (TMB vs Classical). QGD by itself is ok but it's much more fun playing the Nimzo > the exchange QGD, also much better practical chances to play for a win.

There is one free benefit of choosing the Slav which is you get a free answer to the English, although I would not advise choosing it based just on this. It's best to make the decision based on how you feel looking at the positions you get and whether the ideas and plans come naturally. 

I think it also says something that one of the greatest exponents of the Caro-Kann (Anatoly Karpov) played the Slav only twice according to my database, whereas the Nimzo/QID or QGD he played his entire career.

I think it makes sense to decide what it is about the Caro-Kann you like, then make your choice of openings against 1.d4 based on this. For example do you like the Caro-Kann because you like grinding out endgames? Or do you like playing it in a sharp counter-attacking style? If it's the former I would go down the Nimzo/QID or QGD path, whereas with the latter choose one of the sharper Slavs.

I'm a Caro Kann player and use the Avrukh book referred to by the original poster. However, I've never played the Classical Slav but play the Chebanenko. A substantial part of the Avrukh book works well with the Chebanenko, particularly if Black intends to answer 1.d4 d5 2.c4 c6 3.Nf3 Nf6 4.c4 a6 5.e3 with Bf5.
Two points I can't really answer through lack of experience:
- does the Chebanenko route into the Exchange Slav i.e. with a6 included give Black better chances than Avrukh's suggested line ?
- does the mass of theory on the Classical Slav mainline give Black better chances than the murkier lines of the Chebanenko e.g. 5.c5 ?

With respect to the original question, I think the Slav complements the Caro well. I wouldn't discount the Tartakower all together though. I'm really not sure I agree that Black is struggling for equality and would recommend the John Cox book "Declining the Queen's Gambit". The main issue is a satisfactory answer to the Exchange which you are very likely to encounter in club and torunament chess. If you are happy with 1...d5, 2...e6, 3...Be7, thereby facing only Nf3 in the exchange (excluding Nge2) then I think it's a viable choice.

Slav would seem a good fit. Avrukh is a great place to start. Also Sadler's older book has clear explanations.