I want to know whether there is an option to quickly delete eMails from the inbox without sending them to the trash folder in GMail and Yahoo Mail?
No, and probably no…
Gmail doesn't use folders, instead it assigns Labels to messages [comparable to categories].
When you click delete the Email remains in the same folder ("All Mail"). What actually happens is the Label "Inbox" is removed and the label "Bin" is assigned to the message.
The upside is you don't really have to worry about them anymore. All Emails labeled with "Bin" are deleted after 30 days automatically.
For Yahoo I think you can't change the behavior how "deleted Messages" are handled.
On earlier versions of Yahoo Mail you could "tweak" the Spamfilter to directly delete all messages identified as spam. Then instead of deleting an Email you would mark it as Spam and it would be gone.
But even if this is still possible I DON'T SUGGEST to do it. Chances are the Spamfilter will "adopt" to the messages marked as spam and sooner or later all Mails would be considered spam [and therefor deleted automatically without you knowing it]