Is there a way to form a "one who [verb]s" noun?

Question

In English, I can take just about any verb (for example to crush), and form a noun which means "one who [does that verb]" by adding -er (for example, crusher).

Is there a similar formulaic alteration (or set of possible alterations) in German? For example, can I take the verb zerschlagen, and easily change it into a noun meaning "someone who crushes"?

It seems to me that if there is a way to do this in German, there must be a more complicated rule, or more than one depending on the verb. For instance, paint and painter translate to malen and Maler/Malerin, but according to Google Translate sing and singer translate to singen and Sänger.

I am not even sure what to call this -er verb-to-noun construction in English. All my attempts to search on the topic of changing nouns to verbs just bring up information on gerunds, which is definitely not applicable here. (The gerund form of a verb would mean "the act of [doing the verb]", which is definitely not what I am looking for.)

Answer 1

You are talking about agent nouns.

VERB STEM+er does that for many German verbs, technically. But it does not always make sense or sound natural:

läuten → Läuter?
regnen → Regner?
zerschlagen → Zerschläger? or Zerschlager? (some people will refuse the second version because "Schlager" already exists as word for "popular song")

Sometimes, it describes (predominantly) something for doing something:

bohren → Bohrer (drill)
schlagen → Schläger (bat, racket, stick)
mixen → Mixer (kitchen mixer, less frequently used in context with beverages or music)

Furthermore, the transformation does not always describe someone who VERBs:

schauen → Schauer (to look and shower)
füllen → Füller (to fill and filler/pen)
bauen → Bauer (to build and peasant)
lenken → Lenker (to steer or drive and handlebar; "Lenker" actually does have "driver" as second meaning, but mainly in Austria and Switzerland)

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Here are some guidelines the community has come up with so far:

General

• Despite the two counterexamples above, the verb to be transformed should actually describe some acting.
  bleiben, meinen → Bleiber?, Meiner?

• If the verb ends on -eln the Suffix is -ler.
  sammeln, basteln, wickeln → Sammler, Bastler, Wickler

• Pay attention on verbs ending on '-rn'. Their stem is everything but the 'n'. zaubern → Zauberer

• For a female agens, -in is appended.
  Bastler, Mörder, Werfer → Bastlerin, Mörderin, Werferin

Umlauts

• A in the stem often becomes ä if the verb has strong inflection.
  tragen, backen, verraten → Träger, Bäcker, Verräter
  but: beraten → Berater (even though beraten is conjugated the same way as verraten)
  On the other hand, sometimes also weak verbs change a to ä.
  jagen → Jäger

• In its present form, this "rule" cannot be applied to au.
  tauchen (weak), rauben (weak) → Taucher, Räuber
  ^{Although Räuber could be an exception like Sänger.}

• One example for o becoming ö was found, yet its inflection is weak.
  morden → Mörder

• U seems not to change, even with strong verbs: rufen → Rufer

Exceptions

Note that singer was normal German, too (and Meistersinger is still known to contemporary speakers thanks to the popular opera by Wagner). It was replaced by Sänger. So singen does not actually become Sänger, it's just that the two words are so old, they've undergone different developements.

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Another similar but not equal possibility is to use the first participle of the verb as a noun:

zerschlagen → der Zerschlagende

Although you dismissed this possibility, I think there are some cases where it works as you want it to:

der Wissende (one who knows)
der Suchende (one who searches)

Answer 2

In theory you can put the ending -er to any verb, but this ending has a special meaning. If you derive a agent noun X-er from a verb X, a X-er is not just someone or something that is doing X, but whose innate character is doing x. For example, Lehrer (from lehren - to teach) is not just someone who is teaching, but someone whose profession is teaching. Bohrer is not just a thing that drills, but a thing whose sole purpose is to drill. So the ending -er is often used for professions (if persons) and tools (if things). Lehrer refers only to a person whose profession is teaching, Schläger refers to a person who regulary beats others up, Mixer is something designed to mix. It has no other intendet function. Examples like Mörder - murderer might look like they contradict this, cause a murderer might only murder for once, but he will always be a murderer, so in this case -er is nevertheless correct. Zerschläger does not exist in german, cause there is no such thing like a Zerschläger, but if you invent something with the appropriate function, you could call it Zerschläger. Though there is the similar word Zerstörer wich denotes a special type of warships and someone or something that destroys (just like the word destroyer).