1. The word beg only suggests a position of weakness in one context. Although that's how it's most commonly used, it's completely correct (and not uncommon) to use it as a synonym for "ask" if you want to imply a stronger request--check out the second definition in Merriam-Webster for formal evidence if you need it (it's not obscure or archaic). I'm normally all for choosing words based on understandability, but since far more people understand my meaning of the phrase than the "true" meaning, I'd say that the people using the idiom (rather than the words themselves) are the ones who have an ulterior motive other than understandability.
1b. I think it's extremely silly to ask me why I used one word instead of another closely related word. Yes, words have near-synonyms, and I could just ask easily ask you why you're not using my word instead. I know you're trying to imply I have an ulterior motive (that I'm doing it just to be contrarian or to feel right or to win an argument or whatever), but that's a weak way to do it since "I like it better this way" is all the justification anyone needs for using a word that isn't outright misleading or insulting. What you're really saying is that you would make a different stylistic choice, which is not much of an argument at all.
2. The "chip on my shoulder" is only in response to you telling me I'm fighting hard to justify something incorrect, which is quite frankly insulting (and, as I said, an incorrect thing to say in its own right, since the dictionary agrees with me, regardless of whether or not you think I have an ulterior motive). This very specific situation (when someone rebukes someone else because they think the other person is ignorant, when it is they themselves that are ignorant) is a huge pet peeve of mine, and it upsets me when people double-down on that position after being shown their misunderstanding. Of course, they're the ones who have convinced themselves that it is the other person who misunderstood and is doubling down, so that just compounds the insult.