If you want to read more about this structure, it is called the subjunctive. Basically verbs like suggest, recommend, insist, demand, etc. are followed by the verb in the present subjunctive mood. This is identical to the bare infinitive (the infinitive without to). This structure is little used in British English (and, as Dan says, can sound quite pompous). I think it is used more in American English. A British English grammar book will only discuss it briefly in the advanced level book. An American English grammar book might have more detail.

The academic answer is: only A is correct because after "suggest", we use the base form of the verb (not "takes" or "taking"); and C is wrong because "to" is unnecessary (either remove "to"; or change "he to" into "for him to"). But in reality, a lot of native speakers will say B, especially informally. A is a bit posh - if you speak like this around gangsters of inner-city streets, they will think you're fronting and may pick a fight with you. It's considered the proper way of speaking in a formal tone, but this is a nuance that a lot of less educated native speakers don't know, so I wouldn't recommend worrying about it too much. Most native speakers only know it by 'feeling' it after a lot of experience of hearing this kind of sentence in formal contexts.

Because the verb like suggestions, insist, demand are follow only base verb. So the suggested is followed by take.