Finite state transducer with infinitary outputs or without emphasis on acceptance? 1)  Is there a notion of (deterministic) finite state transducer (FST) that allows the possibility of producing an infinite stream of output symbols?  In other words, one where the transduction relation is a subset of $\Sigma^* \times \Gamma^\omega$ (as opposed to $\Sigma^* \times \Gamma^*$)?

For example, the definition of an FST given on Wikipedia involves a transition relation $\delta \subseteq Q \times (\Sigma \cup \{\epsilon\}) \times (\Gamma \cup \{\epsilon\}) \times Q$, which might first seem to permit infinite streams of output.  However, the transduction relation is defined inductively (basically, the reflexive-transitive closure of $\delta$), thereby ruling out infinitary outputs.

2)  Relatedly, the definitions of FSTs that I've seen include a subset $F \subseteq Q$ of accepting states.  Is there a name for the class of (deterministic) FSTs in which $F = Q$, i.e., in which all input strings are accepted and the emphasis is placed on the output?

Any references that you can point me to would be greatly appreciated!
There are several relevant references on the subject. I just indicate a few of them. I suggest you to browse the references of these articles to get more.

Early references include the work of Françoise Gire.

[1] F. Gire, "Relations rationnelles infinitaires", Thèse de 3ème cycle, Paris VII, 1981.

[2] F. Gire, Une Extension aux Mots Infinis de la Notion de Transduction Rationnelle, 6th GI Conf., Lect. Notes in Comp. Sci., Volume 145, 1983, p. 123-139.

[3] F. Gire, M. Nivat, Relations rationnelles infinitaires. [Infinitary rational relations] Calcolo 21 (1984), no. 2, 91--125.

Interesting results can be found in

[4] Latteux, M.; Timmerman, E. Rational $\omega$-transductions. Mathematical foundations of computer science (Banská Bystrica, 1990), 407--415, Lecture Notes in Comput. Sci., 452, Springer, Berlin, 1990.

Section 4.3 of [5] is also a very good reference.

[5] L. Staiger, ω-Languages, Vol 3, Chapter 6 of the Handbook of Formal languages edited by G. Rozenberg and A. Salomaa, (1997) 339-387. Springer-Verlag, Berlin.

Finally, here is a more recent paper, where you will find other relevant references.

[6] Finkel, Olivier. On the accepting power of 2-tape Büchi automata. STACS 2006, 301--312, Lecture Notes in Comput. Sci., 3884, Springer, Berlin, (2006).