Bruno Anglès and Lenny Taelman have published a profound study of finite characteristic special values as listed in the title. This appeared in Proc. London Math. Soc.
(3) 110 (2015) 1000-1032. The analogy with classical theory is clear throughout the paper, as well as some potentially interesting differences; these results bode very well for the future of the subject. I have written a long review of this paper as well as pointing to a very new one on the web. This is been submitted to Math Reviews/MathSciNet and appears here with their explicit permission.
(Added 12-11-2015: A revised version of the paper by Anglès and Tavares Rebeiro cited in the review (along with a third author Tuan Ngo Dac) has been put on the arXiv as arXiv:1511.06209v2. In this new version the result is established in complete generality for all q
A url for a pdf of this review is