I took a ferry between Dover and Calais and had no problem crossing with my bicycle. In fact, it was very easy!
I walked up to the ticket counter with my bicycle beside me, asked if my bicycle was allowed on the ferry, they told me it was but that I had to go to the front of the line of cars and follow the instructions they gave me. I bought the ticket, and rode to the front of the line.
I found myself surrounded by motorcyclists, which they put in the same category as bicycles, and the group of us (approx. a dozen motorcycles and me) were told to go ahead and board the ferry before any of the cars and other vehicles. We were directed to one side of the ferry and told to secure our "bikes" to the piping and whatever else we could find, and then to take what we needed for the trip across with us. Then we were directed to where the lounge etc. was.
We had been told that we would be required to go down to our "bikes" before the rest of the passengers when we docked, so I headed down as directed, unlocked my bicycle and got ready to roll. We were waved off the ferry ahead of all the other vehicles, and sent on our way.
The same thing happened on my crossing between Caen and Portsmouth.
I had a great trip across both times, except for a bit of seasickness at the start of the Dover crossing.
Oh yes, it was recommended to me to head right for the restaurant the moment I boarded the ferry if I wanted to eat during the crossing. Most people stand on deck and watch the White Cliffs disappear and then go to the restaurant, but if you go to the restaurant first, you can just walk right in and get your food ... then watch the White Cliffs disappear. Apparently if you go later, there's usually a huge line-up.