Learn English – She loves being looked at. or She loves looking at her

This is a rather authoritative way of stating categorically that you don't want something to happen generally or habitually, instead of just one time. The key is the progressive - which makes the prohibition somewhat habitual, as if thwarting the onset of a habit or tendency you want no part of.

For example, in the last example, the speaker could be talking to his son, who just got a car and has been home late a few times. Or a daughter that has started dating. Saying just "I don't want you to come home late" could be used for one occasion = tonight, after the prom, etc.

Having said that, the same father could say "I don't want you coming home late" in reference to one occasion, and it is merely a very authoritative restriction, and still carries a meaning of "tonight or any other night."

In (1) and (3) the speaker doesn't want these to become a regular thing; In (2) the speaker wants him to cease ALL inquiries - "stop asking around!" In (4) the speaker doesn't want the fact that the listener has a broomstick to become common knowledge.

Sometimes the gerund and infinitive forms are interchangeable; sometimes they are not. Your examples above ("travelling", "dancing", "getting") are all examples of interchangeable usage. Here are examples where interchanging alters the meaning of the sentence:

"Travelling" vs "to travel":

• He remembered travelling to Canada. (He has traveled to Canada, and now remembers that.)
• He remembered to travel to Canada. (He was supposed to go to Canada, and then he did.)

"Dancing" vs "to dance":

• She stopped dancing. (She never danced again.)
• She stopped to dance. (She interrupted an activity to begin dancing.)

How do you know when the gerund and infinitive are interchangeable? Generally, gerunds are best for talking about completed actions, and infinitives are best for talking about incomplete or future actions.

Gerund: I was getting up earlier last week. (The getting up already heppened.)

Infinitive: I want to get up earlier. (The getting up hasn't happened yet.)

Still, knowing which form to use requires some memorization and intuition. Here is a list of common verbs from http://www.gingersoftware.com/content/grammar-rules/verbs/gerunds-and-infinitives/:

Followed by a gerund:

admit, advise, avoid, be, used to, can’t help, can’t stand, consider, deny, discuss, dislike, end up, enjoy, feel like, finish, forget, get used to, give up, go on, have difficulty, have problems, have trouble, imagine, it’s no use, it’s worthwhile, keep, look forward to, mention, mind, miss, recommend, remember, quit, spend time, stop, suggest, understand, waste time, work at

Followed by either a gerund or an infinitive without causing a change in meaning:

begin, continue, hate, intend, like, love, prefer, start

Followed by a gerund or infinitive but with a change in meaning:

forget, remember, stop

Followed by an infinitive:

afford, agree, appear, arrange, ask, care, decide, demand, expect, fail, forget, hope, learn, manage, mean, offer, plan, prepare, pretend, promise, refuse, remember, seem, stop, volunteer, wait, want, wish

Followed by a noun or pronoun and then by an infinitive:

advise, allow, ask, cause, challenge, command, convince, expect, forbid, force, hire, instruct, invite, order, pay, permit, program, remind, teach, tell, urge, want, warn