Both the grammar and semantics of these constructions can be confusing:
1: allow + to + infinitive: It allows to do something.
This isn't grammatically correct since it is missing a noun phrase before the infinitive phrase.
2: allow + verb+ -ing: It allows doing something.
This is grammatically correct, but semantically it is more appropriate to use it to mean that something isn't prohibited, e.g. City of Victoria allows camping in parks. And, it is more appropriate to use allow something to be done (i.e. passive voice) to mean that something is provided as a capability, e.g. ...allows all records to be sorted. Because even though the entity that is allowed to do something (i.e. an object pronoun) isn't included in the sentence, the thing that is allowed is still to be done by an entity.
3: allow + pronoun + to + infinitive: It allows me to do something.
This is grammatically correct, and probably the most commonly used form.
4: allow + pronoun + verb+ -ing: It allows me doing something.
This is grammatically correct, and equivalent to the second case, although the entity that is allowed to do something is explicitly stated with an object pronoun, which is "me". Therefore, it is more appropriate to use it to mean that something isn't prohibited rather than something is provided as a capability.
5: allow + noun: It allows something,
This is grammatically correct, and can be seen as a generic form of the second case, where the object can be other noun phrases than gerunds.
6: allow + noun + verb: It allows something to be done.
This is grammatically correct, and is usually more appropriate than the second form when it is used to mean something is provided as a capability.