Ubuntu LTS – Is It Binary Compatible with Debian?

Your questions are some what kind of complex so I don't think I can answer them all correctly. If someone read this post and see anything incorrect, please tell me to fix it. And also remind that I can only answer Debian - Ubuntu repo relationship, since I can't catch up Mint development goal and way.

1. Despite the fact that Ubuntu is based on Debian, it only picks the packages that it wants (89% as of 2010) then modifies them (Ubuntu regular stable releases draw from Debian unstable branch "Sid", while Ubuntu LTS releases draw from Debian testing branch, which draws from Sid). So depends on the goal, some Debian packages may be or may not be in Ubuntu. For now, 2015, Ubuntu and Debian are no longer compatible to use each other's repo.

2. No, you are not missing nothing. But also keep in mind that packages number can't say nothing due to packaging method, if you want real comparison, look for source packages number.

3. No exactly. As I mentioned, Ubuntu takes most of its repo from Debian Sid (or unstable, but I prefer the previous name) what is fairly up-to-date. Those softwares get tested, managed for a few (6, I think) months before get released in the schedule of twice a year. In Debian, softwares after got into Sid go to testing (after 10 days if no one can find more bugs). They might not ever go to Debian stable (what would be released) if they don't get to the right time, but if they get to testing in frozen time, they will get tested for another 7 months.

4. That's right, don't never try to do it unless you want to f*k your own system up. Mint is based on Ubuntu and the later 'un is based on Debian, but doesn't means they share the same dependencies. However, there is sth shared along Ubuntu and its derivatives: the ppa.

5. Well, yea, if saying sth like 90-95% then yes.

6. Like Drav Sloan said in his comment, they are not same distros. E.g. wit Debian , you can archive nearly perfect stability, but not in Ubuntu or you can't have attractive installer in Debian.

Debian's unstable branch is and will always be named "Sid", while the other branches (testing, stable, oldstable) are getting a unique codename for each release (Stretch (Debian 9, currently in testing), Jessie (Debian 8, currently stable), Wheezy (Debian 7, currently oldstable), Squeeze...). See this link for more and updated information.

There have been a number of varied boot and service control systems across Unix platforms over its tangled history.

The service\chkconfig based system you found simple and effective is generally referred to as SysVinit style and was a major step along the way to some sort of standardization. You will find this style of boot on RHEL/CentOS(EL) through the 6 release, Fedora through 14, and on Debian/Ubuntu based distributions up until 2015. It was not the only boot system around though, the (simpler) BSD style init system still has many fans.

SysVinit was not a perfect solution (what is?) and Systemd was devised to overcome many of the issues; this is the systemctl command based system you are now experiencing. Although it is not universally liked (people hate change, bloat, etc) there is no doubt that it is fast becoming the defacto standard across the majority of distributions.

Therefore, looking immediately forward the answer to your original question is simply:
The standard way to control services across most Linux distributions is now systemctl!
How long this will hold true is anyone's guess; probably just until something comes along that is better and becomes widely adopted.

I'm sure there will be wrappers available to allow, your current favorite, the service/chkconfig commands to continue to do mostly sane things, but with this particular learning curve it is probably best not to fight it. Maybe looking forward, for a time there'll also be systemctl wrappers for older systems, to make managing them alongside more current ones less of a pain ;)