To begin, a general comment: the treatment of with in ODE is not satisfactory in general. The sense division does not seem to be well thought out, or was perhaps too hastily revised from NODE. Of the 16 senses, at least four are, in my opinion, misassigned: some subsenses should be attached to different core senses; some core senses are actually species of other core senses. Because of the wording of some important senses, it isn't clear in ODE where to place some of the commonest uses of with, e.g., I visited with them last week. Is it 1(1) or 9(7)? Another example of this: ODE gives the following two sentences to illustrate senses that are completely removed from each other: the people shouted with pleasure (at 7(5)) and he was trembling with fear (at 11(7b)). Most people would agree these senses are at least related, if not identical, but ODE has somehow estranged them from each other. In cases where there was no clear category in ODE for placing instances, I have made judgment calls, classifying sentences under the sense that, if better defined, would accommodate them, or in some cases, assigning both applicable senses.
Notes from the spreadsheet:
New sense added (spreadsheet record 6): sentences of the type "he called me with the news," "contact me with the details," etc. have an element of means in them, but are more specific in that the complement gives the reason for the communication undertaken. I subsumed this new sense here, under the main instrument sense, even though there are other with senses that deal with causality (such as ODE 7b and 7c). This sense is found only with verbs of communication.
New sense added (spreadsheet record 18): the sense of with indicating harmony or agreement is not treated in ODE's numbered sense inventory. (e.g., "I'm with you," "You're either with us or against us."). Instead, it occurs in the phrases in ODE, but I think it should be treated as a regular sense because it can have other than a personal name or pronoun as a complement. FrameNet, e.g., has "Mr Attlee was careful to position himself with the majority view in the cabinet." I've placed it under sense 15(9), which shares the idea of concurrence.
Other instances of with that may deserve idiomatic treatment
The phrases in ODE cover the main points. with it, which in informal English can be used attributively ("a really with it teacher") might wreak havoc with the parser.
The informal with PP With any luck typically begins a sentence or clause and acts like a sentence adverb.
Another group of with PPs typically begin sentences or clauses and fulfill the function described in Quirk 9.52, to express accompanying circumstances that may be causal. The example he gives is "With all the noise, she was finding it hard to concentrate." Sentences of this type mostly go under 11(7b) in ODE. Note that these phrases may have important antecedents, e.g., "With all of that going on, I was hardly in a position to buy a new car."
A quick and informal survey suggests that when with ends a sentence (whether a question or a declarative one), the sense involved is often 9(7) Concomitant, and somewhat less often 1(1) Accompaniment, but many other senses are possible.
I notice that the subcorpus labels in the with instances file rarely match the ones shown in the FrameNet instances. There also seems to be a disconnect between the number of records in the instances file and the number of sentences shown by FrameNet in many cases. Don’t know the reason for this.