First of all, a point of terminology. The forms ending in "-ing"in your question are not gerunds. They are present participles. A gerund is a word derived from a verb, that functions as a noun, or in some cases as an adjective. A present participle is an actual verb form. The confusion is that in most verbs, they have the same spelling, and can only be distinguished by their usage.
Secondly, in most of the cases you list, either an -ing form or a present-tense form may be used, with only a slight change of meaning. Consider ther following pairs:
- (1A) He heard the door open.
- (1B) He heard the door opening.
- (2A) Do you like seeing people compete on reality shows?
- (2B) Do you like seeing people competing on reality shows?
- (3A) we could hear the train arrive.
- (3A) we could hear the train arriving.
- (4A) On reality shows, you often see people behave badly.
- (4B) On reality shows, you often see people behaving badly.
In each case, the B form emphasizes the event, while the A member emphasizes the action that takes place. But in each pair, the two sentences are both grammatically valid and have very similar meanings.
The question implies that in each case only one form or the other is correct, or at least that one form is much more widely used than the other in each case, and asks why this is. Well both are correct. It may well be that one form is more commonly used. But usage frequencies follow changes in style and fashion, and are not generally subject to rules.
In addition somewhat similar pairs with no verb of perception involved. For example:
- (6A) The blacksmith had started to forge a sword when I entered the workshop.
- (6B) The blacksmith had started forging a sword when I entered the workshop.
Note that this is a somewhat different form from cases 1-5, and from the examples in the question. Depending on the verb and the construction, in some cases a participle form will be correct, and in some cases it will not. In some cases a non-participle form will be correct. In some either will be correct. My point is that often a sentence using a participle form can be rephrased to use a non-participle form with only minor change of meaning. But not in all cases.