To answer the easy questions first:
- Dos does originally mean dowry. But the word over time came to be used to express something like “excellent quality, particular value,” etc. Lewis & Short defines this as “a gift, endowment, talent, property, quality.”
- Quod being followed by the subjunctive makes it indirect speech – the speech of those who would criticize the fables; in other words, it indicates the reasons they cite for their rejection of the work. You could translate this as: “If someone should want to cavil at this book on the grounds that trees speak etc.”
Now for the difficult question. The phrase quod prudenti vitam consilio monet is dark and has apparently puzzled philologists for centuries. Ørberg offers an explanation, but I am not sure he has solved the riddle.
There seem to be two main schools of thought:
- Those who think that prudenti modifies consilio, which yields the translation: “that it reminds of life with prudent counsel.” And if we note (from Lewis & Short) that monere can also be found “without the accessory notion of reminding or admonishing, in gen., to teach, instruct, tell, inform, point out,” we can say: “that it teaches life” (as if as a discipline, I suppose), or “teaches about life.”
- Those who think that consilio is has no modifier, believe that prudenti describes the recipient of the monition and often emend it to prudentis, yielding the translation: “that it teaches the prudent (or: the prudent man) about life with counsel.” Ørberg seems to be in this camp (minus the emendation). In principle you don't need a dictionary to allow the dative to be used – the dative can be used freely to name beneficiaries. Still, monere has a normal way of naming the recipient, which is very frequently used, and that's the accusative, so the dative is at least a little unexpected. This interpretation also raises the question why the prudent in particular should need advice. (For a spirited rejection of this objection in Latin, see the 1911 dissertation by one Alfred H. W. Tacke titled Phaedriana).
Of course, vitam monere is strange as well. It sounds as if the message of the fables was “Live, love, laugh!” – which as far as I know is not in fact the case. There seems to have been a school of thought that read vita as “those living,” i.e., the writer's contemporaries, yielding the translation: “to teach society with prudent counsel.” This is actually not completely crazy, as this meaning is also attested in other (poetic) contexts (see Lewis & Short). Still, it seems not to have gained much traction.