Here is the answer to your question.

1. Get leads to email: Yes possible. There is an option in the Custom Form Channel called "Send Mail To", just enter the email address where you want to receive the response.
2. Unlimited icons: Bit Assist has a few defaults icon. But you can add the custom icon of your preference.
3. Custom icons: Yes you can add the custom icon of your preference.
4. Custom contact form fields: Yes Custom Contact Form channel is available.
5. Contact form integration with MailChimp & MailPoet: Bit Assist doesn't have integration with MailChimp & MailPoet. Bit Assist supports webhook you can send data to others via webhook. You can use Bit Integrations to get the leads to your MailChimp & MailPoet.
6. Spam protection with Google reCAPTCHA: This feature is not available in Bit Assist's current version.
7. Custom redirection/thank you page after form submission: After form submission, the custom success message is shown inside the widget but does not redirect to any page for now.
8. Google Analytics: Not available Google Analytics all features. Only channel click track with GTM custom event is available.
9. Custom CSS style: You can customize your widget with a custom CSS option. This option is available under the Customizations tab.
10. Custom height position: Yes, this feature is also available.
11. Page targeting: Yes, this feature is available.
12. Country targeting: Not available
13. Available days and hours: Yes this features is available under Setting Tab.
14. Traffic source targeting: Not available

We recomment you please use the free version, before purchsed the pro plan. Most of the feature are aviable in free version.

Here is the download link of free version: https://wordpress.org/plugins/bit-assist/

Check our overview video: https://youtu.be/atVwkzFNnmM
Doc: https://docs.bit-assist.bitapps.pro/