Is it possible to create a form that can be partially filled by our customer and then be completed by us?

I am not sure if this possible at all.

1. I would like my customer to complete a form to start a process.
2. My staff member should then be able to open the form and complete the next phase.
3. Management should then open the form to sign it of. 

Same goes for leafe forms, Employee complete the form, thereafter management open the form to approve or decline the leave.

Yes, this is possible.

1.1 Create the form with all the fields you need
1.2 Hide the fields that you don't want the customer to see (How to Hide Form Fields)
2.1 By default, email notifications will include an Edit Link that allows the form owner to Edit the Submission data.

Or you can do this from the Submissions Page. (How to Edit Submissions Made on Your Forms)

2.2 Using the Get Form Page URL Widget you can show the hidden fields in Edit Mode only.
Complete Guide: How to Show or Hide Fields on Edit URL

You can also send an email when the staff member completes this phase.
(How to Send Email Based on User's Answer)

3.1 If you want, you can also implement an Approval Process.
Complete Guide: A General Approach on Building an Approval Process Workflow