An analysis of the grammatical sentence Looked after carefully, this coat will keep you warm through many winters will explain why Not sticked to the rule, the employer will/would discharge you is ungrammatical.
Looked after carefully ... is a reduced participle clause. It replaces the full passive equivalent If it is looked after carefully ... . The active voice equivalent, as noted by the OP, is If you look after it carefully, ... .
In both the active and passive unreduced sentences the pronoun it is in the initial (dependent) clause, and anticipates coat, its antecedent and the subject of the main clause.
The pronoun it has been ellipted from the reduced clause looked after carefully, but it is there implicitly and anticipates the antecedent coat in the main clause in the same way.
When the implicit it (or indeed any pronoun) in a reduced clause does not anticipate or co-refer to the subject of the main clause that follows it, then we have what is called a dangling modifier (or dangling participle, or misrelated modifier, etc). And ambiguity is often the result.
Peters in The Cambridge Guide to English Usage in the section called dangling participles (p138) notes the various terms for the same usage, and states:
Whatever term is used, the grammatical problem is how an independent
introductory phrase stands in relation to the rest of the sentence. In
Latin grammar it is no problem because inflections mark the
independence of such phrases. English lacks inflections to do this,
and so the introductory phrase will seem modify the subject that
follows it, with strange consequences for the meaning sometimes.
Now damaged in the stern, the captain ordered the ship back to port.
Garner's Modern American Usage (Danglers, p221) makes a similar point about the subject of the main clause following the dangler:
Most danglers are ungrammatical. In the normal word order, a
participial phrase beginning a sentence should be directly followed by
the noun acting as the subject in the main clause.
If this is not the case, "the sentence becomes illogical or misleading."
A misleading sentence can be constructed from the OP's example by changing the subject of the main clause:
Looked after carefully, you can wear this coat for many years.
Now we have a basis for analysing the OP's ungrammatical sentence: Not sticked to the rule, the employer will/would discharge you. Correcting the participle, and omitting would results in:
Not stuck to the rule, the employer will discharge you.
which can be interpreted as intending to mean If you don't stick to the rule, the employer will discharge you.
The problem is that Not stuck to the rule in the original sentence is not a feasible reduced passive clause. The expression needed here is to stick to the rule containing the phrasal verb to stick to. An exemplification of this is She doesn't stick to the rule.
The passive version of this exemplification is The rule is not stuck to by her. This could conceivably, although rather improbably, be reduced, to The rule not stuck to (but not Not stuck to the rule as in the OP's example). Removing the noun gives us Not stuck to, which could form the start of a sentence that continues with the consequence of not sticking to the rule:
Not stuck to, the rule will fall into disrepute.
A more plausible example of the same construction is:
Not seen to, the wound will become infected.
Note that in both cases the subject of the main clause (the rule / the wound), immediately follows the participle clause, whose implicit subject is identical or co-referential. We therefore have sentences that meet Garner's criterion for grammaticality and Peters' for clarity.
So, in summary, the sentence Not stuck to the rule, the employer will discharge you does not work.
Firstly, because it is not a correct passivization and reduction of the clause If you don't stick to the rule, which is my interpretation of what the OP intends to say.
And secondly, because even if it were rewritten as a correct reduction, not stuck to, its implicit subject (the rule) does not match the subject of the main clause. It is hence a dangling modifier.