Hello,
I believe this is because of the particular data you entered for your contacts.
But let's start from the beginning. First of all, if there's any problem with filling in a field, i.e. it's empty or it doesn't include the record you were looking to put there, it's because the record doesn't contain required values in certain fields.
As you probably noticed, when creating a project, the only company you could assign is the one you set group to client. This is because that particular field in Projects requires Companies with group set to client.
Now the tricky bit is to know what are the criteria used for each field. In EPESI it's pretty easy to check that. All you have to do is to start adding/editing a record and hover your mouse over the label of field in question:
These particular fields you have problem with are limited to Employees, i.e. Contacts assigned to the Main Company. Usually Main Company is the one created during installation, but you can switch the Main Company in administrator settings (you should only do this when absolutely necessary, it's best to have it set once and not to tinker with it later). Contacts assigned to Main Company are considered "Employees" by EPESI, granting basic employee rights and include them in various employee-only fields.
Hope this helps,
Arek