It sounds like you may have the Broadband Access ability on your plan which gives you the ability to use your bberry as a modem.
You can cancel that if you wish but you always have to have a data plan for blackberry.
Email will be included with your data plan and shouldn't give any extra charges, unless they are being sent as text messages.
I use gmail and my company account on mine with no issues, and I used to have my yahoo on there also at one time. I never received excess email charges
The text message inbox is just a generic inbox it handles all messages SMS, MMS or email.
You can separate them by going to your message folder- Blackberry Button> Options>General Options>SMS and EMail Inboxes- Change to Separate. Then Exit and Save.
You will then be able to see what is truely going where.
If you yahoo is going to TXT messages then it is probably a setting on Yahoo.