There are agent nouns for all genders.
For example, saltare gives rise to saltator, saltatrix, and saltatrum.
For more details, see this question.
The stem is revealed by the genitive form.
For my three examples they are saltator- (third conjugation), saltatric- (third), and saltatr- (second).
(The stem of rex is reg-, so it has a g instead of a c.)
If you ...
Yes, this is attested in Classical Latin, particularly in the case of the non-human serving as an agent (taking the preposition). Allen and Greenough, §405:
The ablative of the agent is commonest with nouns denoting persons, but it occurs also with names of things or qualities when these are conceived as performing an action and so are partly or wholly ...
The following is my summary of Silvia Luraghi 2010 paper, in the tabular format (obviously, here I summarized those parts that are relevant to your question only). All the examples are hers, including the translation.
Luraghi 1986 writes that in Latin
“human agents are usually marked by a prepositional phrase introduced by the preposition ab, which takes ...
Are there any existing diminutives of agent (-tor/-trix) nouns?
Yes, though the rarest.
Examples for -trix have been already mentioned by @Joonas and @cnread: nutricula, meretricula...
I'd like to add an example for an adjective derived from a -tor word: punctatoriola, as the diminutive of punctatorius from punctator: the reference is Festi Fragm. e Cod. ...
Ovid's Remédia Amórés 422 suggests that your grammar is mistaken and that non-humans with agency can be considered agents:
Á cane nón mágnó sæpe tenétur aper.
Cicero's Dé Officiís 1.68 suggests that, at least metaphorically, emotions can be considered agents:
Nón est autem cónsentáneum, quí metú non frangátur, eum frangí cupiditáte, nec quí invictum ...
I think there is a semantic difference between "a Gnaeo" for the agent and "sica" for the instrument. The agent will be expressed in this way only in a passive sentence, but the ablative of instrument can be used also in an active sentence (Marcum sica occidit). The agent is normally animate. Note that there is a difference between "animate" (=humans, gods, ...
I can see three somewhat different meanings here, and they all lead to different choices.
If you want to say "he is carried in a litter", then the litter just describes location and is not really an agent.
In this case you would use in with ablative and get in lectica portatur.
If you want to say "he is carried by a litter", ...
Not only can animals and non-humans personified act as agent, but humans can act as instruments, as we see in Livy III.37:
Et decemvirí, quí prímó tribuniciós hominés, quia id populáre habebátur, circum sé ostentáverant plebí, patriciís iuvenibus sæpserant latera.