I run Fiddler 24/7 for weeks and have never seen any problem like the one you describe.
All of the errors you mention are errors that would be shown when the browser's DNS lookups fail; browsers do not use DNS when Fiddler or FiddlerCore are capturing their traffic, so your question is especially confusing.
Is there some reason you believe this is related to HTTPS-- does the problem not occur if you don't enable decryption, for instance?
In the failing scenario, if you go to a command line and type nslookup google.com
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