Learn English – perceive as + verb ing

This question has previously been asked and answered on ELU. To summarise, there are three different syntactic constructions for the verb allow...

1: With a gerund complement indicating what is allowed:
Mama don't allow no drumming (non-standard English for Mama doesn't allow [any] drumming)
Nor does she allow smoking reefers (perfectly valid example of a gerund complement)

This construction does not normally accept a patient/object (the person/thing being allowed to do something). You sometimes see things like "She doesn't allow JJ Cale smoking reefers", but most native speakers do not like that much (it should be "She does not allow JJ Cale to smoke reefers", per #3 below).

2: With a noun phrase object indicating what is allowed:
Some states do not allow abortion
This school allows the use of computers during exams

3: With a noun phrase patient/object and an infinitive complement (what they're being allowed to do):
He won't allow me to leave
Some states allow automatic weapons to be kept at home

It is worth noting that last example, showing that the "patient" direct object (who/what is being allowed to do something) doesn't need to be a person (in this case, it's automatic weapons).

It's also worth noting that you may see things like "The iPad allows to surf the Internet on the move", with no "patient" object. These are not considered grammatical (they're usually from non-native speakers).

(In case it is not obvious from the above, OP's #1 is ungrammatical. All the others are fine.)

I cannot find any specific questions about the closely-related verb let on either ELL or ELU, but I'll just make one final point here. Be aware that syntactically, let doesn't work exactly the same as allow.

Option 1 is incorrect.

Option 2 is correct. This example uses a "split infinitive", which I think is OK. (Some people disagree with me about that.) The example would be even better with an article (or other determiner) before "user". For example:

A user can also opt to not provide shear reinforcement

or

The user can also opt to not provide shear reinforcement

The "to <verb>" phrase is English's version of an infinitive. It is often used after a helper verb, like "opt".

It is unusual to have "to <verb>ing" in a sentence. One such situation is when the word "to" is used link gerunds. For example, "I wanted to speed up, so I switched from walking to running."