Yes, you can do this.
I think you pay taxes the same way as for your US customers, but hopefully someone else can confirm.
Yes, an LLC can do business in Canada. You should probably be aware of the major taxes that you might need to pay, namely the GST/HST and the income tax. Read CRA's exposition for non-resident businesses.
The GST/HST is a value-added tax on goods and services. You might need to collect and remit this tax to the Canada Revenue Agency, depending on the type of goods/services provided, how much work is actually done in Canada, and whether your annual revenue exceeds $30000. The GST/HST is not LLC-specific.
Your LLC may be subject to Canadian income tax, depending on how much work is actually done in Canada. Even if you elect to treat the LLC as a flow-through entity for income tax purposes, Canada still treats the LLC as a corporation, and the LLC may need to file a Canadian income tax return as a non-resident corporation.