We will be staying in Copenhagen 5 nights starting Oct 19. Do we need reservations?
Probably. Use Booking.com to see what's available for your dates. You can do lots of refining of the search down the left side of the screen to get the numbers down for browsing.
In my experience, you can always find something without reservations. Copenhagen is a central hub for business conferences (except in summer vacation season), so lodging can be expensive whether you book ahead or wait. In October, I'd try to save money by finding lodging (b&b or hostel) that doesn't cater to the business traveler - but it really depends on what you want/need from your lodging.
Since you know you will be in Copenhagen, when, and for how long, I don't see any advantage to not having reservations. Of course, if the above dates are just an estimate (if you're driving around, say, and may stay longer or shorter in places before Copenhagen), that's different.
One of the great advantages of having the internet available, it gives you a chance to research and check out hotels/hostels/pensions ahead of time. I hate wasting time when I arrive trying to find lodging when I can be checked in and then out sightseeing!
As was said, if you're time is set, you might as well get reservations. We normally book only first night, last night and any dates we are locked into - but try to keep those to a minimum. Most of the time we travel without a set itinerary and without reservations. I do what Ken suggests and research lodging options online ahead of time. I create an excel spreadsheet with relevant information (price, rating, address, phone number) for about five places (more for destinations that may be crowded) and store it in my Ipod. I call ahead mid-morning before I arrive. With very few exceptions I tend to end up in my top one or two choices and normally at a better price than the price I researched.