Let $\zeta_K(s)$ be the Dedekind zeta function for a number field $K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/conjectured about the next term?
On a related note, the BSD conjecture predicts the value of the first non-vanishing Taylor coefficient of the Hasse-Weil $L$-function of (say) an elliptic curve. Are there any conjectures about the coefficients after that?
There is a related conjecture for the L-function attached to certain Galois representations, due to Colmez and stated in the beautiful survey paper Periods by Kontsevich and Zagier (see section 3.6). This survey paper refers to an Annals paper of Colmez: Périodes des variétés abéliennes à multiplication complexe (Jstor link), and to some papers of Hiroyuki Yoshida.
There is also a book by Hiroyuki Yoshida called Absolute CM-periods, where some of this might be discussed.
I have no idea whether this is related or not to the Rössler-Maillot/Kudla story mentioned above in the comments.
The conjecture as stated by Kontsevich and Zagier is for a Galois representation sending complex conjugation to minus the identity matrix. This is a rather restrictive condition, and Kontsevich-Zagier also state (without explanation) that "in general, one does not expect any interesting number-theoretic property for subleading coefficients".
The questions about Birch-SwinnertonDyer are much subtler than the first question, and I do not pretend to have anything to say about it.
Edit: and, indeed, the following bits of information are a "weak" answer, at the level of saying "yes, just as the Euler-Mascheroni constant (and a family of such constants) appears in zeta, similar constants provably appear in the Laurent expansion at 1 for Dedekind zetas." This is very distinct from any discussion at the midpoint of the critical strip, I agree, despite class numbers' appearance at 1. (Some potential confusion about whether that central point is 1 or 1/2, due to traditional normalization of zeta functions of elliptic curves.) [end-of-edit]
About the first question, it has been known for some decades (though I do not know a citation, perhaps because not so much came out of such ideas) that, following Shintani (and generalizations by Satake decades after), especially for totally real fields (where the discussion was motivated by special-value results at positive even integers, as an approach complementing Siegel's and Klingen's), the non-zero ideals can be expressed as a finite sum over ideal classes, each of which can be expressed as a sum over elements of a lattice modulo units, ... the key point being that the latter has (many) reasonable sets of representatives from the intersection of a "rational cone" with a lattice. For very general reasons (extrapolated in Ash-Mumford-Rapoport-Tai, but anticipated before... e.g., by Shintani) this intersection of lattice with rational cone is a finite sum of sums of positive-integer-coefficiented sums of lattice points...
Shintani's goal, and Satake's, was to obtain an expression for values of L-functions with a meaning a bit different from Siegel's or Klingen's (not quite addressing things like Lichtenbaum's K-theoretic conjectures, but... who knows?)
Incidental to that, as I myself once considered (fruitlessly), the simple classical argument about Laurent series of zeta at $1$ extends to give an analogous, obviously fussier, result for totally real, and probably other, number fields.
I think the analogous question for all other (automorphic) L-functions is much subtler. Edit-edit: e.g., consider the Kronecker limit formula! (Again, not midpoint of critical strip, but edge...)
