Suggested answer:
(1) Based on the corpus results, the statement is only true of begin, bother, and continue. Examples:
:... then we can begin doing things.
I begin to avoid Cordelia.
Please don't bother lying.
Don't bother to answer.
It had taken all his strength to continue acting the stupid dullard...
...she will continue to act as if she and her daughter are one...

Claim and offer occur only with the to-infinitive, e.g.:
... he claims to be only thirty-two.
We seem to appeal to the miraculous...
The police offer to stay with him.

Postpone only occurs with verbs ending in -ing, e.g.:
'Maybe if they'd postponed living together till they were married, she'd be alive now.

(2) The quickest way to check whether any form of the verbs combine with either -ing or to is to perform two searches per verb, e.g. bother* *ing and bother* to; this secures all verb forms of bother followed by either -ing or to. Then we need to check the results to see if we are dealing with a verb ending in -ing or the to-infinitive and not some other word form.

(3) Yes, all three verbs seem to have the same meaning regardless of whether they are complemented by -ing or to; see examples under (1).

(4) When stop is followed by a verb in -ing it suggests that something has stopped or is about to stop (i.e. you have/is about to quit doing something, while when followed by the to-infinitive it suggests that you stopped in order to do something else. E.g.:
Could you stop staring at me like that...
They did not stop to ask her any questions.

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Read about verb + -ing or to-infinitive here and here
Terminology: -ing participle clause, to-infinitive