[RPG] How to prevent players from using Persuasion or Deception to weasel their way out of a murder

There are many ways to gain additional skill proficiencies in the rules.

If you or your players want to make a new set of proficiencies, you're free to do so using a customized background: (PHB 125)

You might want to tweak some of the features of a background so it better fits your character or the campaign setting. To customize a background, you can replace one feature with any other one, choose any two skills, and choose a total of two tool proficiencies or languages from the sample backgrounds.

The Skilled feat (PHB 170) is self-explanatory:

You gain proficiency in any combination of three skills or tools of your choice.

As a DM, you can let your players get training in skills in order to gain proficiency: (DMG 231)

A character might be offered special training in lieu of a financial reward. This kind of training isn't widely available and thus is highly desirable ... Possible training benefits include ...The character gains proficiency in a skill.

However, consider that this bonus is in the same list as gaining an extra feat, so it's not something to be done lightly.

Finally, both the rogue and the bard have ways of boosting their skills. Rogues can double their proficiency bonus for a handful of skills through the Expertise feature (PHB 96) and bards get half of their proficiency bonus to any check that doesn't include their proficiency bonus (PHB 54).

You might be interested in the Background Proficiency variant rule

DMG 264 has the "Background Proficiency" variant rule, which seems to hew pretty closely to the mechanics you desire:

With this variant rule, characters don't have skill or tool proficiencies. Anything that would grant the character a skill or tool proficiency provides no benefit. Instead, a character can add his or her proficiency bonus to any ability check to which the character's prior training and experience (reflected in the character's background) reasonably applies. The DM is the ultimate judge of whether the character's background applies.

You don't need specific rules to make crazy situations

Why did the designers get rid of the skill point system in the first place? Mike Mearls said:

We unified the progression for skills and weapons under one set of rules. That removed a lot of complexity and allowed us to include skills in the simplest version of the game without adding a lot of rules overhead.

Therefore, instead of depending on an arguably unwieldy and complicated set of rules for skills, you can use the simplified framework of 5e to create the kinds of specific "proficiencies" that you describe in your question.

More specifically, you can let your players add their proficiency bonus to specific kinds of checks, or even give advantage for more significant bonuses. If your character is an expert in astrology, maybe let them add their proficiency bonus to Arcana, Nature, or Religion checks that involve the stars. Examples of this kind of bonus exist throughout the system: for instance, the Belt of Dwarvenkind gives advantage on persuasion checks against dwarves.

Indeed, such cases are what the advantage mechanic was made for: the DM giving quick bonuses in response to favorable situations. You might find that this goes a lot smoother than a skill point based system. As a player, I've found that DM bonuses like this are not only balanced (because they're highly situational), but also a lot easier to keep track of than, say, 3.5's super complicated skill system.

Without nudging your players into particular actions, there are many ways you can set Droki up so the characters can have the best chance to catch him. I will enumerate them broadly and leave you to the idea-generation:

Droki is unattentive in the Whorlstone tunnels

Inside the Whorlstone Tunnels, however, the Derro is distracted by ever-present threats, giving the party the best chance to catch him.
[...]
Droki's route takes him all around the Whorlstone Tunnels, and he's too scatterbrained to notice anyone following him. (Out of the Abyss, p. 68 & 69)

The characters could catch Droki by surprise while they follow him around the tunnels.

Use Random Encounters

The Whorlstone Tunnels aren't empty, and Droki is as likely to find random encounters than the players. Think up a scenario where Droki is interacting with one of the Random Encounters in the dungeon where the players can get the opportunity to snatch him up. What I personally used was:

The random encounter with the Flumph. The Flumph knows much about the cultist and their activities, so it's only natural that it knows about Droki as well. I let the Flumph tell the PCs that Droki follows a specific route and hardly deviates from it. The Flumph can then point them to where he will go next and the PCs can setup an ambush.

Lastly, capturing Droki is optional, their main goal is to:

... follow Droki, see what he does and where he goes, and report back. Or if they see an opportunity, seize him and either bring him in for questioning, or kill him and bring back evidence of his activities. (Out of the Abyss, p. 63 "Search for Droki")

If your players have lost interest in actually catching him, don't force them into it. Give them the evidence they need to bring back to Errde (when they earn it, of course) and that's that.