The functional difference between adverbs and adjectives is what they are qualifying or describing, respectively. That is, an adjective gives further information about the associated noun or noun phrase. An adverb, in contrast, qualifies not only verbs (as the name indicates), but also adjectives, other adverbs and even whole clauses.
As you guessed, the word which is modified in your context is the verb to choose. The adverb tells that you do not need to choose a particular weight, but instead can just blindly grab for any one weight. Or, rather, choose arbitrarily.
There is no clean way to parse "More important, ..." strictly as written, but the intent is easy to understand.
A critical point is in the answer you reference:
...the adverbial phrase ‘more importantly’ modifies nothing in the sentence. What is wanted in constructions of this kind is ‘more important,’ an ellipsis of the phrase ‘what is more important.’
(Note that this is a subject with some disagreement, and the above quote expresses one opinion (of many) in the debate.)
If we accept that "More important, ..." is short for "What is more important is that..." then there is no issue at all: important modifies the fact expressed in the that clause.
Consider the following similar sentences:
Finding the treasure is important.
It is important that we find the treasure. (using an expletive it)
If we ask "What is important?" the answer is "that we find the treasure". (Or, "finding the treasure" in the first setence.)
You can also read "More important, ..." simply as:
This fact is more important: ...
"More importantly, ..." doesn't usually modify a verb in the main clause:
We lost the the treasure. More importantly, we lost our friends in the woods.
You didn't importantly lose your friends -- that doesn't make too much sense. Rather, you lost your friends, and that fact is more important than the fact about the treasure.
Best Answer
Actually, in this example, patronizing is the adjective. Sometimes is (and always will be) an adverb. Allow me to explain.
Patronizing is the present participle form of the verb patronize. Using the present participle turns a verb into an adjective. In this case, patronizing is being used as an adjective to describe the portrayal of women.
Sometimes is being used to modify patronizing. So what part of speech is it? Well, it is being used to modify patronizing, which, even though it serves the function of an adjective, is still a verb by nature. A word that modifies a verb is an adverb.
So we can come to this conclusion from two directions: