are mobile phones a business write-off.
Yes, as long as you use it for business purposes. If you use the phone partly for business, and partly for personal use, then you can only deduct the business portion (but you already knew that). As an example, if you only use your cell for business 75% of the time, then you can only deduct 75% of your expenses related to the cell.
I wouldn't be surprised if there were loopholes, but I think the IRS is actually pretty clear on this aspect of the law. The problem arises in determining what percentage of the time you use it for business versus personal, and then in proving that figure if you ever get audited. Short of keeping a log every time you use it for personal versus business, you are most likely going to end up with an estimate. (Looking at your phone records should help.) If it were me I would err on the side of caution. If you think you use your cell 75% of the time for business, then maybe only deduct 70%. You should consult an accountant about this.
For more on this see IRS Publication 535 Business Expenses. On page 3, under Personal versus Business Expenses, Pub 535 states:
Generally, you cannot deduct personal, living, or family expenses. However, if you have an expense for something that is used partly for business and partly for personal purposes, divide the total cost between the business and personal parts. You can deduct the business part.
must my account be designated as a business account?
I don't know, but if it were me I would designate it a business account. It's the safer way to go, and as Hondo69 pointed out, it will help your business establish a credit history, which is important.