I always thought not pronouncing the /h/ had to do with informality until I took an Americna English course where teachers would make sure we omit the sound on pronouns such as him, her.
a) What about the last sentence I wrote in my original post? Are there specific rules as to when you can omit the /h/?
Not pronouncing the /h/ in words such as him, her, his, himself, has, have isn't wrong at all. By leaving the /h/ out in these words, it is easier for the speaker to keep the rhythm and flow of speech. So, by saying /im/ instead of /him/ is quite normal. Those forms are called weak forms (like /f?/ instead of /fo:/ for the word 'for').
The initial letter 'h' isn't pronounced in three cas
In American English, you leave it out if it's not stressed or it's not at the beginning:
Her car will be sold to a Japanese guy. She'll seller car to a Japanese guy. (= sell her)
In non-rhotic accents like BrE the situation might be different. In those accents, I am afraid if you leave out the aitch, "her" will sound the same as "a", which is weir
To make things even more complicated, it is possible that in very rapid speech, the schwa sound may be used for: a, are, I, her, of, or and have. But normally the context helps to define the meaning.