What is the fee to integrate authorize.net and Merchant account with Apptivo?

No, there are no additional transaction fees incurred when you integrate authorize.net.

You link your authorize.net account using the steps provided in Authorize.Net payment gateway

As far as your merchant account is concerned, you only need to link your merchant account to your authorize.net account. There is no additional setup involved. To summarize:

1. Open your authorize.net payment gateway account. 
2. You get your merchant account from any financial institution of your choice.
3. Link your merchant account to authorize.net account. 
4. Provide your authorize.net information in the Invoices App.