@PiotrAchinger: My notes say that Proposition 3.1 of K. Coombes "The $K$-cohomology of Enriques surfaces", Contemp. Math 126, (1992) proves this for Chow motives with $\mathbb{Z}[1/2]$-coefficients, though I haven't had time to double check. There is also some generalisation (but only with $\mathbb{Q}$-coefficients ) in the recent paper D. Kawabe, "Chow motives of genus one fibrations", Manuscripta Math. 175 (2024).

See the top of page 216 in S. Zucker, Generalized intermediate Jacobians and the theorem on normal functions, Invent. Math. 33 (1976), no.3, 185–222. Note that Zucker points to Theorem 16.16 in P. Griffiths, On the periods of certain rational integrals. II, Ann. of Math. (2) 90 (1969), 496–541. Griffiths in turn says this is due to M. Rosenlicht, Generalized Jacobian varieties, Ann. of Math. (2) 59 (1954), 505–530.

Yes I think that’s right, Sreekantan’s article only looks at the case $q-2a\geq 1$. This is because the regulator he constructs basically comes from the boundary map in the localisation sequence. I don’t know a reference for a similar style conjecture when $q-2a=0$ I’m afraid.

You could try F. Charles “On the zero locus of normal functions and the étale Abel-Jacobi map”, and M. Green, P. Griffiths, K. Paranjape, “Cycles over fields of transcendence degree one”, and the references contained therein.