No, they are not fully interchangeable. Perhaps the best way to illustrate is to ring all the changes on the various words of your list and the different examples you give us to explore their distinctions:
1) A person of diverse interests can talk on many subjects.
A person of various interests can talk on many subjects.
This is a valid substitution; it differs from the "diverse" version only in that "diverse" implies and stresses a greater range and variability in the interests. The person with "various" interests may be able to talk on many subjects, but the person with "diverse" interests can talk on many subjects which have surprisingly little to do with one another.
A person of different interests can talk on many subjects.
Used this way, it is somewhat non-idiomatic, raising the question "different from what?" "Different" doesn't imply "many", merely "not the same".
A person of distinct interests can talk on many subjects.
Similarly, "distinct" does not imply "many", merely "clearly distinguishable (from each other)". But used this way, it doesn't mean that either; it would be read as an alternative definition which leaves out the "from each other": "marked" or "notable". Thus this sentence, which is also somewhat non-idiomatic, interestingly suggests something quite distinct from the previous: it's not the number of interests that allows a person to talk on many subjects, but how readily you can distinguish his interests as his.
A person of disparate interests can talk on many subjects.
"Disparate" implies "far apart (in some characteristic)". It implies even more strongly than "diverse" that we're referring to someone who has many interests, but it goes even further. It describes the interests not merely a many, but as having nothing to do with one another. It would actually succeed in communicating that the interests were "distinct" (in the first sense) in the way using "distinct" wouldn't. This sentence says that someone who has many interests which are non-overlapping can talk on many subjects.
2) We have looked at various houses, but have decided to buy this one.
We have looked at diverse houses, but have decided to buy this one.
The houses we looked at were in a wide variety of styles.
We have looked at different houses, but have decided to buy this one.
Identical meaning to the original.
We have looked at distinct houses, but have decided to buy this one.
Nonsensical. Houses are typically readily distinguishable from each other and everything else.
We have looked at disparate houses, but have decided to buy this one.
Not very idiomatic. We looked at houses which had nothing in common -- we have no idea why the agent showed them to us, she seemed to be picking them at random, so we're firing her and getting a new one.
3) The department store sells many different things.
The department store sells many diverse things.
It's remarkable the breadth of the department store's inventory.
The department store sells many various things.
Almost identical to the original, but doesn't emphasize the variety of things sold as much as "different".
The department store sells many distinct things.
Nonsensical.
The department store sells many disparate things.
The store has the most random and ecclectic inventory of any store I've been in.
4) The criminal was charged with two distinct offenses, one being murder, the other bribery.
The criminal was charged with two diverse offenses, one being murder, the other bribery.
Nonsensical. If there's only two, how can they be diverse? Diverse implies a group of things across a range.
The criminal was charged with two various offenses, one being murder, the other bribery.
Nonsensical. Again, this word implies "many", and if there's only two, it doesn't make sense.
The criminal was charged with two different offenses, one being murder, the other bribery.
Perfectly fine, but unlike the original, doesn't emphasize that the two charges are being distinguished from one another. Perhaps in the "different" formulation, they were two separate criminal instances, a month apart. But in "distinct", there's the implication that he could have been charged under a single criminal statute, but the state decided to break it down into these two. Perhaps murder + bribery is a bad example: consider,
After the star witness was found beaten into a coma, the criminal was charged with two distinct offenses, one being attempted murder, the other witness intimidation.
The criminal was charged with two disparate offenses, one being murder, the other bribery.
In the criminal justice system, when evaluating offenders for likelihood to reoffend, one thing which is noted is "criminal versatility", which refers to the diversity of types of crimes for which the inmate has been sentenced over his or her criminal career; inmates with high levels of criminal versatility are more likely to reoffend.
That is the only circumstance I know of in which the disparateness of a criminal's crimes might be under discussion. "Yeah, he's only had three arrests, but for one of them, he was charged with two quite disparate offenses, one being murder, the other bribery. So there's more evidence of criminal versatility in his case than it seems at first blush."
5) Don't confuse these disparate concepts.
Don't confuse these diverse concepts.
There's a lot of these concepts and they cover a lot of ground.
Don't confuse these various concepts.
There's a lot of these concepts. Try to keep track.
Don't confuse these different concepts.
They may look the same, so I understand how you might confuse them, but actually these concepts aren't the same, so don't confuse them.
Don't confuse these distinct concepts.
These concepts quite clearly do not overlap, so you really don't have an excuse for getting confused by them.
I hope that helps!