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Removing the contact won't remove messages or entries from the call logs, because those messages/calls were still sent/made. However if you have deleted the contact I'd expect the name to disappear from those entries (the number will of course remain). If it doesn't, try rebooting your phone.
To remove entries from your call log, go to your call log, press and hold on the entry you wish to delete, select "delete" from the options.
As for text messages, just delete the messages or the whole conversation.
I have deleted the whole conversation for 1 of my contacts however, when I click on the recents when trying to send a picture to someone, if I click on any of the contacts info and then click and hold the little button that pops up once I choose one, the complete history for that contact pops up, including deleted text messages. Seems like this would use up unnecessary memory? and I find it hard to believe that there isn't a way to do away with/disable "Recents" altogether.
Which app is that in? I've never seen that button thing you ringed in my decade of using Android, so I'm guessing it's a feature of the particular app (which may be your manufacturer's phone app - the phone app is the place I assumed your "recents" list was found).