"Expenses" are usually business-related costs you make (like taking people out to lunch, having to buy a decent suit, etc). Such costs are generally covered by an "expense account" (a little portion of extra money set aside for you, apart from your regular salary, to cover your expenses). Indeed, in this context, often used in plural; like: "I get my usual fee, plus expenses."

You are correct. "Expense" is just the singular of "expenses". A singular "expense" would refer to some item that can be separated from other cost. My plane ticket was an expense. All the items together would be expenses. However, "expense" can also be used as a verb. If I "expense" something, that means that I can submit it in an "expense report" to my company and they will pay me back. Usually a single expense on the report is referred to as an "expense item" or an item on the expense report. "I had to buy some equipment for the job and payed for it myself." "Oh, that's all right. You can expense it."