"This is considered taxable income and is only to be used for moving expenses"
No. As taxable income, it can be used for anything. As Opus said in one of your other questions, they just increased your pay. That is the same thing for the "tax assistance". It is just a way of 'grossing up' your pay to cover expenses.
https://ttlc.intuit.com/questions/4469382-due-to-relocation-expenses-included-as-part-of-my-gross-wa...
https://ttlc.intuit.com/questions/4470562-why-is-relocation-tax-assistance-taxable-and-is-included-i...
Because your employer is just treating it as increased wages, you don't count it as a moving expense reimbursement. So include the full amount of moving expenses.
Write a contract with a receipt showing you are renting the apartment for storage of your things, and it can be includes as moving expenses.
Paying your relatives to help you move is a "gray area". If the majority of their time on the trip is directly related to moving, I would probably deduct it (assuming the cost would be less than paying a moving company). If they also spent a significant amount of time doing non-moving things (such as sight-seeing or other vacation-type things), then probably most or none of that would be deductible.
