I'll answer your second and third questions.
In the example provided, working is a gerund, a noun, as is work. Those words do not functions as verbs below, which is why to work doesn't fit:
 We start working tomorrow. <gerund>
 We start work tomorrow. <noun>
 We start to work tomorrow. <to-infinitive>
 We start back to work tomorrow. <to-preposition>
Want to is a set phrase (e.g., supposed to, have to, going to), which is why -ing doesn't fit:
 I want to go there.
 I want going there.
 I want to going there.
To requires a base form (i.e., to go), not an inflected form (i.e., *to going).
Does that help so far?
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