What is your Update Layout setting? What is your meter setting? I certainly wouldn't think a half rest after one beat is correct either.

 "Allow dotted rests" is turned on ... 

What happens if you turn this off?

As Ernie has indicated, you have not given us enough information to know what is happening.

https://makemusic.zendesk.com/hc/en-us/community/posts/215729367-General-Discussion-House-Rules 

Sigh.  Now even I can't replicate the problem, even with all the same settings.  It now copies correctly.  I'm curious why it wouldn't do it before.

Automatic update layout is on.  4/4 meter.  
The operation was to highlight the first two measures shown (both staves) and drag/drop them into the last two measures, which were previously empty.
These are two measures from a much taller score.  I was not using any type of Stack copy.
Previously, the result had been the same if I used Ctrl-click to paste the measures, and also the same if I used Ctrl-C/Ctrl-V to copy and paste.

I certainly wouldn't think a half rest after one beat is correct either.
I'm not sure why this is being brought up.  It pasted a dotted half rest.

If I turn off Allow Dotted Rests, it now (still) copies correctly.  [Of course, I would like to leave this setting as on, since there are a lot of dotted eighth notes in the score and I'd like to be able to duplicate those as-is instead of having them turn into eighth+sixteenth rests.]

This whole situation isn't that big a deal with just two measures, but I'm concerned about it happening if I do a huge cut/paste.  Going through a large section to manually get rid of dotted half rests in 4/4 is not appealing!

Now even I can't replicate the problem, even with all the same settings. 

Jerry, I think that we have all been there. Yet another Finale Mystery (cue spooky music).

[spooky music continues]

Shaggy:  Zoinks, Scoob!  Let's see who the culprit REALLY is:

[Fred yanks off mask]

Velma:  Jinkies!  It's the Bank Street Music Writer from Commodore 64!

I certainly wouldn't think a half rest after one beat is correct either.
I'm not sure why this is being brought up. It pasted a dotted half rest.

The thought that no matter the dotted rest setting a half rest on the second beat wouldn’t be good. IMHO, of course.

I feel your pain. Very frequently, copying and pasting measures (especially if I start the copy from a portion of one measure) will mess up the rests. Rather than filling in a 4/4 measure with three quarter rests or a quarter + half note rest after a quarter note that was part of the paste, I get a melange of smaller-value rests. Selecting all and hitting Retranscribe (under MIDI/Audio) fixes it. But Finale should be smarter than this, and it's never been able to do this in all the decades I've used it. I'm not sensing any major bug fixes are forthcoming, judging from the responses I've gotten from tech support about bugs that have been reported over the past 1-2 years, so I just continue to use Retranscribe when needed. Good luck!

But Finale should be smarter than this, 

I can't imagine that any of us disagree.

Both MuseGroup and AVID have promised AI integration in the next versions of MuseScore and Sibelius. Dorico has claimed an AI component since 2017 (though most users doubt its existence). Will the owners of MakeMusic make a similar investment and, more importantly, will it do any good?

I'm going to be optimistic and say "yes and yes." Given the inertia many of us have commented upon for some time now, that might not be reality-based.