assuming the portion of my allowance that is not used on such expenses is taxable,
Sorry, it's all taxable. Only the part used to pay qualified tuition expenses would be non-taxable, but you won't have those as a post-doc.
If you have conference expenses that are not reimbursed, you can deduct them as work-related expenses on form 2106 or schedule C, depending on how you file**. And if you pay insurance premiums, you pay with taxable dollars but are then eligible for a premium tax credit ("Obamacare") and/or to deduct the premiums as itemized deductions on schedule A. For a typical single person, you would have to pay pretty high premiums and have pretty high expenses to get any tax benefit, so you should expect to pay tax on all your allowance.
You also need to figure out how you will be paid. At my large university, post-docs are in the same payroll system as everyone else and get a W-2 like everyone else. The important difference is that because post-docs are technically still students, they don't pay social security or medicare tax on their wages, but they do have federal and state tax taken out of their pay.
The other option is that you will receive all your pay without any deductions and will get a 1099-MISC instead of a W-2. The fellowship should be reported on box 3, not box 7, which is for self-employment ("work" -- you are a student, not a worker) (even though your post doc wouldn't last long if you didn't "work" in the common sense, it is not "work" in the tax sense)". If you are going to get a 1099-MISC and not have withholding, then you need to pay estimated tax to the IRS 4 times a year to avoid a penalty for underpayment.
**Now, here's a complication. If you get paid on a W-2, you can take non-reimbursed work expenses as an itemized deduction subject to the 2% exclusion. This means you get a reduced deduction or no deduction, depending on your other income and deductions. If you are single and don't own a home, you probably won't be able to actually deduct your travel. If you are paid on a 1099-MISC in box 3, then you technically don't have income from working and so you can't deduct work-related travel. BUT, if you were self-employed ("working" for compensation) and file using a schedule C, then you can deduct your health insurance premiums and travel expenses "off the top." But, you would then owe income tax AND 15% self-employment tax on your net earnings.
If you are paid on a 1099-MISC instead of a W-2, it is conceivable that it would be more favorable to file as self-employed instead of as a student. But still, probably not. That 15% self employment tax is a big hit.
My university provided health insurance benefits through the payroll system and they were tax-advantaged that way, and my travel was always covered by my advisor. It sounds like you are at a small school that doesn't have a lot of fellowships (or maybe a large school that has decided to disadvantage the students in order to save on paperwork.)
Sorry for the bad news.
