The subject is latus. Definition 6 in OLD is most relevant here:
6 (of solid objects, usu. w. abl.) To be bathed or soaked (in a fluid specified or implied), run, stream, overflow, etc.)
For comparison, there's Ovid Metamorphoses 9.57-58:
vix tamen inserui sudore fluentia multo
bracchia, vix solvi duros a corpore nexus.
...arms streaming (with) ...
One can split up the process of finding the case to three steps:
Find all possible cases a word could possibly be. Also bear in mind that there might be several options for the base word, like supplici coming from either supplex or supplicium. Check the declension tables if you don't remember them by heart.
Analyze the grammatical context. Does the word go ...
As L&S put it, in their classic textwall style (entry for in, II.C.2):
Of the object or end in view, regarded also as the motive of action or effect: “non te in me illiberalem, sed me in se neglegentem putabit,” Cic. Fam. 13, 1, 16: “neglegentior in patrem,” Just. 32, 3, 1: “in quem omnes intenderat curas,” Curt. 3, 1, 21: “quos ardere in proelia vidi,...
Let me make some remarks on what you say above: "Imagine you want to say something like "from us to you [plural]" (where "from" indicates ablative and "to" dative). Since the order is usually uninformative in Latin, nobis vobis is not precise enough. Would something like a nobis vobis be enough?".
As pointed out by Joonas, context is important here. For ...
I agree with the other answers: though ambiguity sometimes is inevitable, the ablative wouldn't be used alone in this context. Here is an example from Plautus that almost exactly parallels your case (with some previous lines added for context):
Gel. [...] Quid igitur me volt?
Croc. Tritici modios decem rogare, opinor [te volt].
Gel. Mene, ut ab sese ...
Ambiguity like this is commonplace in Latin.
For example, "we have to help you" can be nobis vobis auxiliandum est, where the two datives happily mix the two roles.
(In this specific case one of the datives can be replaced by an agent a nobis, but sometimes ambiguity is inevitable.)
Even though Latin word order is flexible, it does contain information.
A Latin adjective can sometimes be read either as a mere attribute or more broadly.
For example, consider these two translations:
Homo conscius intelligit.
1. A conscious man understands.
2. A man, being conscious, understands.
In the first translation conscius is a mere attribute, describing what kind of a man is in question.
In the second one there ...
Despite its name, the Latin “ablative” is not normally used on its own for motion from a place or person. (Often, as in nobis and vobis, the “ablative” is historically not an ablative at all, but an old instrumental). I would stick with “a nobis vobis”.
As for your 1st question, meo judicio is clearly ablative (cf. also meā sententiā). And, yes, you're right: this use is often referred to in Latin grammars as "Ablative of specification".
As for your 2nd question, the particle quidem has recently been analyzed as a marker of emphatic affirmative polarity. This issue seems to be more complex than I expected (...
The upgraded malo sentence mentioned by Hugh has 4 ablative nouns in a row since the first malo is a verb and two of the others are adjectives.
I've got 9:
Ea re, concitato equo periculis imminentibus, ipso illo die hora nona, Flumentana porta spe duce Roma profectus sum.
"For that reason, having spurred the horse in the face of the imminent dangers, that ...
Is this an example?
Cicero: sibi enim bene gestae, mihi conservatae rei publicae dat testimonium.
Perhaps it can be argued that sibi and mihi are datives of reference, but "agent" seems most natural to me. "He testifies that he performed good deeds, but that I preserved the republic." Unless a dative of reference is usual with testimonium, which I don't ...
Francesco Cavalli called the mass he wrote in 1675 Missa pro defunctis per octo vocibus and that's still the name by which we know it. No doubt it should read "Missa pro defunctis octo vocibus", where octo vocibus is in the dative.