Each verb needs to be considered individually. Context and accepted usage are the things that count, not abstract rules.
The
context needs to be considered individually. It would be onerous if one had to consider each verb individually - though I understand that you mean the verb in its context.
But, it doesn't change the fact that we use the present progressive for things that are happening now, and that is universal.
"He's having a good time". "You're being naughty". "He's mentioning something".
"She's wishing she had made a different decision". "He's hoping to get an A".
I think this is the point kadioguy is asking about by stressing the word 'now". If it is, there's only one rule, and all verbs follow it.
Me: "I hope I win".
You: (for some reason)"He's hoping [that] he wins". "He hopes he will win". All verbs can be used in the first way in the context in which the original question was put.
'Mention' does not belong your class of verbs like hear, want, like.
Nevertheless, in the right context, you can say "I'm liking it", "He's wanting the kids to be quiet", "I'm hearing noises in my head".
Also, you can use the simple present with any verb as well. "He mentions the assignment".
To understand the principle, you need to read more, and note the tenses used in context. There's only so much generalising we can do from here.