First, subject or object do not enter into this in any way. It looks like you're looking for nouns corresponding semantically to verbs. Once you have the nouns, they can be subject or object.
There is no such one to one mapping. Verbs that describe actions or states can have corresponding nouns that describe those actions or states as concepts. However, each of these can describe different perspectives of those verbs. For example:
to run = a run (one act of running), running (the act of running)
to kill = a kill (one act of killing), a killing / killing (the act of killing)
to teach = teaching (the activity of teaching), *a teach (one act of teaching - does not exist other than in idioms like 'a teach in')
However, these can work in different ways through idioms and metaphoric extension. So you have sentences like:
- She made a killing on the stock market.
- The killing of civilians is unacceptable.
- The sniper made 5 kills that day.
There are many other suffixes used for various other semantic purposes: translate > translation, agree > agreement, etc.
Your examples, teach > teacher, and kill > killer make actors out of the verbs. This is a very productive method that works on any English verb (even though you won't always get an idiomatically straightforward result, e.g. inspector v. inspecter).
If you have agent nouns, you can also have patient nouns. E.g. the taught, the killed. Or a special, more limited, case train > trainer > trainee but not teach > teacher > *teachee. There are a slew of other suffixes used for Latinate roots. E.g. officiate > officiant, translate > translator, etc.
It could also be that you're actually looking for morphological derivatives of verbs which are called deverbatives or deverbals. These can be either nouns or adjectives, and in English, they're simply the participle or infinitive forms of the verb used as nouns or adjective. That is very simple both morphologically and syntactically. But you need to be careful about the semantics. There's also a subtle semantic and syntactic distinction between sentences like 'the running of a company is hard' and 'running a company is hard'. But that would probably just complicate things.
This article will show you some of the complexities.