"I trust that you give me an answer tomorrow."
Translation:Jag litar på att du ger mig ett svar i morgon.
Non-native here, but I think it's moreso the difference between "about that" or "on that" (literal meaning) and "like" or "as that" or "as to." Your sentences could be then translated "it seems as though she is tired" and "I like that you are coming tomorrow" or perhaps "I am thinking about you coming tomorrow" - not really sure if that is grammatically correct Swedish, either. I don't think the direct object affects which you use. On another note, I think you should say "jag tycker om att du kommer i morgon."
No, the V2 does not apply in subordinate clauses (and not in questions either). In subordinate clauses, the subject goes before the verb, and if there's an inte, that also goes before the verb (this is called the BIFF rule – 'bisats inte före finita verbet').
If you want to say I like that you are coming tomorrow, the word order in Swedish would be Jag tycker om att du kommer imorgon. The main clause is
Jag tycker om [subclause]
In the subclause, the subject goes before the verb. Sometimes the verb still ends up in second place, sometimes not, that just doesn't matter in subclauses. But if you add inte everywhere, the difference gets clearer:
Jag tycker INTE om att du INTE kommer imorgon
– after the verb in the main clause, to preserve V2; before the verb in the subclause, because of the BIFF rule.
You're right, both structures are good but the latter is more common. It's a matter of information structure. If you put i morgon first, it will be a sentence about what is going to happen tomorrow. If you put du first, it will be a sentence about what 'you' will do tomorrow. There are more situations in life that call for the latter.
It is a common variant of i morgon, and I think the only time the distinction matters is if you want to stress that it's "tomorrow" (rather than stressing some other part of the sentence). Then you'd probably use the space (only a guess), since in speech you would generally say it slower and more clearly - but otherwise they're completely interchangeable.