Regarding doubt, how do we apply Romans 14 in difficult situations?

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I find Anchor Yale Bible Commentary (AYBC) on Rom 14:23 quite helpful.

"But the one who has doubts is already condemned if he eats." The "weak" Christian, who has not yet acquired the inner liberty of the "strong," would be led astray by following the example of the "strong," because he or she would really be acting against the dictates of conscience. If weak Christians were to eat meat or drink wine to spare themselves the criticism or scorn of the strong, then they suffer condemnation (lit. "have been condemned," the pf. tense is used to express the condition in which they find themselves).

because the eating does not proceed from conviction. The basis of any moral act is the dictate of conscience. If to eat meat were not governed by the dictate of conscience, then sin would be involved.

For whatever proceeds not from conviction is sin. Lit., "all (Gr: pan) that is not of conviction (Gr: pistis) is sin (Gr: hamartia)". Paul ends his discussion with a maximlike utterance. The statement encounters three difficulties: 1. How generic is pan? 2. What is the sense of pistis? 3. What is the sense of hamartia?

Augustine (...) understood pan as applies to everything, "in every case." Augustine was applying Paul's dictum to the controversy with the Pelagians and was really going beyond Paul's meaning. By contrast, John Chrysostom (...) rightly understood pan to refer to all such indifferent matters as those instanced in the preceding context (vv 2-3,5): all dietary and calendaric observances.

As Augustine understood pistis, it would refer to basic Christian faith, as in 1:17 and 3:25,28. He took it thus in his controversy with the Pelagian Julian, maintaining that all deeds of pagans prior to justification were sinful (...). Again, he went beyond Paul's context, in which it is a question of Christian "weak" and "strong", not of pagans. But with other medieval commentators, Thomas Aquinas (...) understood pistis as meaning "conscience" ("omne quod est contra conscientiam"; ...) ... This interpretation of pistis as conscientia has been attributed to Origen (...), Ambrosiaster (...), and other patristic writers, who use the word "conscience," but it is debated whether these early writers really meant the same thing as the philosophical idea of conscientia. .... Luther (...) realized that pistis could be interpreted "in a double sense," as "opinion and conscience," but preferred "the absolute sense in the fashion of the apostle, as identical with faith in Christ." So too .... Yet many modern commentators (...) have realized the problem that this interpretation creates, in that it disregards the neutrality of many human acts, and therefore understood pistis to mean "conviction," a meaning acknowledged by BAGD (Greek-English Lexicon of the New Testament and Other Early Christian Literature), i.e. a confidence that proceeds from Christian faith, which is distinct from it and manifests itself as liberty with regard to indifferent matters (adiaphora) as in vv 1b, 22, 23a. [emphasis mine]

Hamartia would denote not the power controlling human beings (3:9b) or indwelling sin (7:17), but the basic idea of hamartanein, "miss the mark": the Christian would miss the mark by conduct in a specific case that proceeds not from inner freedom or from a conviction about the act to be done.

Applying the above exegesis, the AYBC interpretation (by Joseph Fitzmyer) is:

Clearly, one has to respect the full Pauline context of 14:1-15:13 and the utterance in v23, maximlike though it sounds, should not be made into an absolute. The "maxim" is not applicable to all Christian conduct as such; it has to do with "all" indifferent matters, such as eating meat, drinking wine, and observing feasts. It is not a rule governing the basic relation of a Christian believer to God, much less the conduct of unbelieving pagan. See ... [another commentary].

Finally, here's the NLT (New Living Translation) of Rom 14:23:

But if you have doubts about whether or not you should eat something, you are sinning if you go ahead and do it. For you are not following your convictions. If you do anything you believe is not right, you are sinning.

So to answer your question

How do we interpret the text, in light of the difficulties which the normal interpretation creates? And to do so without making the text to mean nothing at all?

I agree with the AYBC reading of the Rom 14:23b, that the meaning of "all" should be controlled by vv2-3,5 and even more proximate, by 14:23a. For the whole chapter 14 Paul is talking only about food (which for believer is a matter of indifference, since what is unclean is what goes out, not what goes in), so why isolate Rom 14:23b from the rest of the chapter and over-generalize it?
The examples you cited (police offer, poem, doubting personality, marriage, handgun) are doubts created by situations, not by doubts whether Jesus is Lord, not by doubts whether you have been saved, not by doubts whether the eating meat is OK ("strong" Christians don't have this doubt). I would place it in the category as how to choose your investments in 401(K): stocks, bonds, cash. Do you really want to apply Rom 14:23 to this? For those situations, prudence is needed; prudence informed by both practical experience of the world, by counsel of good friends, by your intellect applying faith in Jesus, aided by the whispering of the Holy Spirit. Don't you see how decisions of those kinds are completely different in kind than what makes you believe Jesus in the first place?
I like NLT's translation of Rom 14:23b: "If you do anything you believe is not right, you are sinning". Look at this translation logically: If X ("do anything you believe is not right") then Y ("you are sinning"). Therefore, if you KNOW for certain that doing X is wrong, and yet you do it, X is true, then Y follows. But in all your examples, you have doubts. That means you cannot decide. That means X is false. That means the "If X then Y" rule does NOT apply.