I'm using Rainlendar 2.

I added a todo last month which is to pay my electricity bill. Of course, it's recurring monthly so I set up its recurrence. However, after I paid the bill last month, I marked it "Completed" and did not show up again this month.

How do I mark a task done for this month and get it to appear again next month?

Of course, I didn't have to add it to my todo list every time I receive the bill, this is what the recurrence feature is for. I already tried marking it "Not Started" but apparently, it shows that I didn't pay my bills last month.

Right now, I just added it to my Events list to get it to occur next month, but in the Events list, you cannot mark it "Completed".

If you set a recurring task completed it will start to show the next instance of it immediately so you actually don't see the completed-state for the tasks. If the task defines also the start date in addition to the due date the task won't be shown in the to do list until the start date for the next instance has passed. You can make these kind of tasks visible if you uncheck the "Hide upcoming tasks" from the context menu.

How about the alarm?

I set it two days before the due date. Will it alarm again next month?

Yes.