If I create a sheet called tab1_MyBets, do I still need the MyBets sheet, or do I have to create MyBets sheets for each tab?
You don't need a sheet named MyBets, but you do need a separate sheet for each tab. Assuming your worksheets are named tab1, tab2 and tab3 you need 3 worksheets named tab1_MyBets, tab2_MyBets and tab3_MyBets.
There is no option on the Excel Log Prices dialog box to specify where the current bets should go - is that automatic?
Yes, it's automatic as long as you have named the sheets correctly as explained above.
Same with the results. Can I split out the results for one tab to go to one worksheet - I see there is the ability to specify which results sheet I log to but is that dependent on each tab being for a different market?
You must select a separate results worksheet for each tab. If you choose the same worksheet on different tabs then the results will overwrite each other.