According to the above, when {-'egh} and {-chuq} are used there is a
subject, but how is it possible that there isn't an object as well ?
In the {-'egh} case isn't the subject the object as well ? And in the
{-chuq} case isn't the other party the object ?

The text says the no-object prefix must be used with the reflexive suffixes. It doesn't say anything about whether there can be an object. One might reasonably conclude that requiring a no-object prefix also implies no object, but it doesn't actually say that.

Couldn't we use the prefix trick with {-'egh} and {-chuq} ?

Aside from breaking the rule about using only no-object prefixes with the reflexive suffixes, I don't see any problem with it. I have recently speculated that the requirement to use no-object prefixes may not be about restricting which prefixes may be used with reflexive suffixes but about telling the reader which prefixes typically make sense with them.

"Okay, I want to say We see ourselves. See is legh, reflexive is -'egh, and we is maH. What's the right prefix? Lemme look at the chart... Hey, there's no we–us prefix listed! What do I do? Oh, it says here to use the no-object prefix with reflexive suffixes."

So, why couldn't we use the prefix trick with the {-'egh} and {-chuq} ?

Mostly because of the rule that says they only allow no-object prefixes. But we have one canon violation of that rule, in paq'batlh: quv HIja'chuq Don't speak to me of honor! Some people question the strength of this as an example, though, because ja'chuq appears in the dictionary as a lexicalized word, so this may be an example of a root+prefix that has fossilized into its own root.

I suggest avoiding playing tricks with the reflexive verb's prefixes, simply because it's so unclear whether it's allowed or sensible.

-- 
SuStel
http://trimboli.name