Can I Create a Form With Branching Logic and Integration with Salesforce and Box?

The answer is - Yes.
FormTitan is a powerful platform that enables you to create the smartest forms using many of its capabilities, including branching logic, and integrations with 3rd party applications. It also goes without saying that these features can be combined together in harmony - all in the same form.

What is Conditional Logic and how to use it
Conditional  Logic enables you to create dynamic forms, that change their route according to the filler's input. So instead of using a static form and having to ask all fillers to answer all the questions in it, you can add conditions that display different questions to different fillers.
Using field conditions you can easily show or hide fields, turn them to mandatory or non mandatory, make them enabled or disabled, Set values in them etc.
Using form conditions you can define what happens when the form is submitted. You can have it redirected to a different URL, you can display a message or send a custom email or push data to a 3rd party.
Read about the conditional logic

How do I integrate with Box and Salesforce?
FormTitan provides over 750 integrations with 3rd party applications and platform. Box, is of course among them.
Read more about our Push integrations
As for a Salesforce integration - we have the most powerful, seamless, bi-directional integration with Salesforce, that allows you to Write as well as Read data. It enables you to connect to as many SF objects as you wish (regular or custom), an also do this with a hierarchical relationship (parent-child). 
Read more about Salesforce Push and Salesforce Get