"How do I reflect those expenses on my 1040? The expenses were not handled through the company, therefore not on K-1, nor can I really be considered an employee and therefore deduct unreimbursed expenses on Schedule A. Can I properly deduct these expenses on Schedule E in the non-passive income/loss from partnerships section in Part II."

You are correct unreimbursed partnership expenses are taken on the Schedule E in Part II. It states that right in the instructions to the schedule E. A partner is not an employee so you will never have anything the Form 2106 for unreimburese partnership expenses. This will also help reduce your SE tax.

The partnership aggreement must have something in writing stating which expenses will not be reimbursed to take them on the Schedule E, if you want to be in complete compliance with the law and the IRS ever questions it.