In reply on your questions:
Can I have brandable (put my own logo) control panels in VPS, Reseller or Dedicated
Yes, for VPS and Dedicated plans. On reseller one, you have a default theme without any brands which is not customizable.
When you say "a few dozen of simple sites like just a couple of html pages", do you mean I can have for example 40 different small html sites (~5 pages) with 40 different Control Panel
Control panel can be only one on the server. But if you mean 40 different accounts for users in it, yes, you can have that (keep in mind, Plesk licensing costs depend on the amount of accounts the license is for).
Are the control panels I will make for my client on all three plans similar?
Sorry, I'm not sure I got you here. Can you please elaborate on this question.
What do you mean by medium popularity? How many people usually enter medium popularity sites?
It is hard to define "medium popularity site" as server load and resource usage depend not only on the amount of site visitors but also on how well the site handles the load. Using that term I wanted to say that hosting a site with hundreds of concurrent users whose requests go to multiple scripts (like popular forums for example) in VPS is almost impossible.
Lunarpages for sure manages the reseller server and I (customer) should manage the dedicated, right? How about VPS?
VPS management is your responsibility like on dedicated plan as well.
I do have a reasonable amount of Linux experience as it is my primary OS, If I can change the server settings, Am I responsible for the security of the server? Who takes care of the server security?
Since server managed is your responsibility on both VPS and dedicated plans, the security of your server is also your responsibility as nobody else can know how the server is setup to secure it.
What I made a wrong setting in the VPS server and it went down, will lunarpages help me bringing it up?
Yeah, sure. We will be helping with issues like this.
I hope this helps. Please advise if you have any further questions or concerns.