Sentence c) is grammatical because 'dummy' there forms a single chain with its 'associate' NP, four men. The chain minimally has two 'links' in this case (more if you consider the hidden structure between Comp V and Spec Tense, but that is inmaterial now). Any chain, whether it is constituted by a 'dummy' there and its NP associate, as in your c) example, or by a displaced DP and its 'trace(s)' (= its 'copies', under the 'Copy Theory of Movement'), as in your b) example, needs 1) a valued theta role feature (a 'Theme' one, here, supplied by the unaccusative verb arrived to its complement four men), 2) a valued Case feature (nominative, here, supplied to the 'dummy' link of the chain by the Tense element visible in the finite verb *arrivED), and 3) a set of valued 'AGR' features - [Number: Plural], here - supplied to the chain by the 'associate' NP link four men and inherited by 'dummy' there - which inherently has no valued Number feature and so is compatible with both singular and plural NP associates. Furthermore, if the chain is the only argument that can become subject of the sentence and satisfy EPP, as in this case, 4) one of its links, the NP/DP or its associated 'dummy', must be inserted/land in Spec T/Infl to satisfy EPP. As in your c) example the two links of the chain {There-four men} jointly satisfy all such conditions, the derivation complies with all the output constraints established at the interfaces and the sentence is grammatical.
The same happens in your sentence b), except that in b) it is the NP four men itself that directly raises to Spec T, receives a valued nominative Case from T there, satisfying the Case Filter, and thereby also satisfies the EPP constraint at the same time.
In your sentence a), on the contrary, the chain has a single link, the NP/DP four men, which contributes 1) a valued Theme role feature, and 2) a valued plural number AGR feature, but, as it remains in situ and, there being no 'dummy' there associate, the subject position (Spec T) remains empty, the chain neither acquires a valued (Nominative) Case feature nor satisfies the EPP. With the 'Case Filter' violated and the EPP unsatisfied, of course, derivation a) 'crashes' at the interfaces and the corresponding sentence * Arrived four men is ungrammatical.