1. No. They are design steps. The result of each design step is a schema.
2. No. That sounds like a homework exercise.
However, at the risk of violating the "homework help only to those who show effort" policy, I will say that the order you list the design steps is exactly backwards. Conceptual design occurs when you are examining the problem space and simply capturing the entities of interest, and how they relate to each other. Logical design is when you examine those entities and relationships, and determine what about them is interesting enough to capture data about. Physical design is when you translate your logical design into something you can store and manipulate (usually on a computer), along with concessions to performance, ease of use, security, etc.
Yes, security is considered to be at the physical level. Once you have created an actual database, you grant permissions as to what logins can view, update or add, delete. You have to plan how security policies will be implemented...do I use Microsoft-style "trusted" connections? Individual logins? Service accounts to talk to the client layer, and let the application deal with its own permissions?
I read the article behind the link you posted, and it might be that you are confusing artifact with activity. For example, the ACT of "conceptual design" produces the ARTIFACT "conceptual schema". A schema can be textual or it can be a diagram. In many cases it will be a combination of the two...a picture supported by text details describing what the picture represents. Just to muddy things up more, the ARTIFACT "schema" is also sometimes referred to as "the design".
As you progress through the stages of design the level of your documents' detail grows. One project I worked on for a major financial institution wound up with a logical schema of around 600 entities and about 1,500 pages of supporting text. Physical design was correspondingly large. Not to brag, but the project was successfully implemented, has been in production for over 10 years and has never had a significant outage. Okay, yes I'll brag. I'm very proud of that system. :)