To as a preposition and not as a preposition

[Alexandrenglish]

Can you please, tell me how I can explain why PLAY doesn’t bring an ‘s’ in the sentence below?

* Do you need a ticket to see your mother play at the theatre?

I know I’m not supposed to :? :? add an ‘s’ to it, but I don’t know how to explain such a thing.

Can you also tell me when ‘to’ is considered a preposition and when it is not? In the sentences below, ‘to’ is considered a preposition and that’s why any verb which follows it is supposed to have ING added to its end but in ‘I need an axe to cut some branches off’ or ‘I want you to know the truth’ ‘to’ is not considered a preposition. Why not?

Sentence 1: In other words, to understand your nightmares may be the key to stopping them.

Sentence 2: What step is Peter taking today to getting a new job?

I appreciate your help.

Best regards,

Alexandre Luiz Borges

[Milanya]

Do you need a ticket to see your mother play at the theatre?

In this sentence "play" is an infinitive, infinitives do not require "s."
It does not have a particle "to" in front of it because it is a bare infinitive.

BARE INFINITIVE. An INFINITIVE without to (win rather than to win), used:
(1) After modal verbs: I must go.
(2) In the pattern verb of perception plus object plus infinitive: We heard the door bang.
(3) With some verbs: Let go; Help me do this; Make them pay.
(4) After rather than and sooner than: I'll go without rather than pay so much.
(5) In cleft sentences: All I did was ask.

Can you also tell me when ‘to’ is considered a preposition and when it is not? In the sentences below, ‘to’ is considered a preposition and that’s why any verb which follows it is supposed to have ING added to its end (not true) but in ‘I need an axe to cut some branches off’ or ‘I want you to know the truth’ ‘to’ is not considered a preposition. Why not? "To cut" and "to know" are infinitives.

Sentence 1: In other words, to understand your nightmares may be the key to stopping them. ("to" is a part of infinitive "to undestand").
Sentence 2: What step is Peter taking today to get (not getting) a new job? ("to" is a part of infinitive "to get").

I am going to school. "To" is a preposition of direction.
You can look it up in a dictionary: http://www.thefreedictionary.com/to
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con·text - The part of a text or statement that surrounds a particular word or passage and determines its meaning.