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...)