Questions about authorship rank and academic politics
I should point out one of the most famous of co-authors in math: Paul Erdős. Erdős never won a fields medal or even wrote a seminal paper as a solo author. What he did do was "wander" around to different universities and help people solve problems. He was an extremely prolific writer (over 1500 published manuscripts) and is considered one of the greatest mathematicians of the 20th century. The ultimate mathematical Renaissance man who could seemingly work on any problem. I think you would have quite a bit of difficulty in arguing that because Erdős never wrote a classical mathematics paper as a solo author, he was not a tremendous mathematician. (Though to be fair, in the past decade, his paper "On Random Graphs" can now be considered a classical work of mathematics. It just took 50 years for the work to become highly applicable).
I relate this story because having lots of co-authors does not necessarily mean having bad skills. Now I am not saying your professor is the second coming of Paul Erdős (I can't think of anyone currently in academia who can claim that mantle) but they might have other skills you are not giving them their due credit for.
Never underestimate the power of being able to organize a group of smart people and get them to work together on the same problem.
To answer your questions specifically-
It's neither normal or not normal. Tenure positions are rare enough that there are certain metrics that are highly predicative (e.g. individuals who are ahead of the publication curve for their field tend to get hired in tenure positions) but there is no conclusive metric. Individually, there is enough variance within each position that you cannot really say.
Collaborate! Having only solo author publications is probably not wise (unless you are in a field that is big on monographs). Having a variety of publications where you are first author, second author, and a middling author show your ability to collaborate and be a team player.
Yes, clout does increase likelihood of getting accepted. Being a big wig with 10,000 citations at an elite university means you are more likely to get a grant than someone with a handful of publications from an R2. But to be fair, the qualities that led that person to be a big wig probably translate into their writing and research caliber as well.
I have never seen any research or evidence that suggests that.
I think you are misinterpreting the data. What you seem to think of as "wise political decisions" is really just collaboration. Collaboration is a good thing in general. I'm not sure why you think that the leaders of the field would put up with someone just "hanging onto their coattails". I think they would probably resent that implication.
My suggestion is that you give up trying to "stand above everyone" and find some people who are as good as you are and start to share ideas and start some collaborations. If you wind up in a place that has a small faculty, then having already established a circle of collaborators will help your career greatly.
The person (maybe people) you open with have good positions because they were judged to have good potential. Their "citation counts" and all of that were just evidence of that potential. Their vast network of collaborators was likewise.
As you say, you are in a field in which insight is (all) important, but also rare. Sharing ideas can help. Hoarding doesn't.
I'm not sure what part of theoretical CS you're considering. In the parts I'm familiar with, co-authored papers are very much the norm, and author lists are alphabetical, so there's no significance in being first or last author. Other areas use other author-ordering criteria but co-authored papers still seem to be the most common case.
Citation counts differ dramatically between areas. For example, I have just finished a 50-page (co-authored) paper that cites about 25 other papers. In contrast, a seminar announcement on our departmental mailing list linked to a 14-page paper in another area of CS that cited more than 80 papers (or, should I say a 9-page paper with a five-page bibliography?). If that paper is typical of its field, citation counts must be much higher there than in my area. Perhaps your citations-vs-single author comparison is comparing a field where a typical paper has one author and cites a lot with a field where a typical paper has several authors and cites less.
There are plenty of universities where co-advisorship of PhD theses is common. Think of it as a sort of insurance policy in case the student doesn't get along with one of their advisors, coupled with two heads being better than one. It's not a reflection on either advisor's skills as an advisor.
I guess that the coauthorship was earned for some implementation work. [...] I may be envy but I find very little proofs of that scientist actual skills. Instead I have a feeling that the scientist makes very wise political decisions or is good as self-advertising.
Wow, you're spectacularly dismissive. From reading your question, I get the impression that you believe that scientific publication is primarily to prove how awesome the author is, that co-authorship must necessarily dilute that awesomeness, and that the only reason to bring on a co-author is to get them to do the boring parts. That's not how it works. Furthermore, unless you're reading papers that explicitly state what the contribution of each author was, your guesses of who did what are exactly that: guesses, and guesses based on almost no information. Don't condemn people based only on your baseless guesses.
c) Are indeed funding and promotions based on citation counts? If this is not the case, how those researchers without solo papers are assessed?
No. Not applicable.
d) Is it true that those researcher who has successful solo papers are much better supervisors?
Why would it be? They seem completely unrelated concepts, to me. If I had to guess, I'd guess that the researcher with many co-authors is better at working with other people, so is better at working with students. Doesn't that make more sense?
I am talking here about theoretical research in CS/Applied Math. I guess in other fields there are many more opportunities for equally valuable contributions of many sides.
Your guesses about theoretical CS aren't something that I, a theoretical computer scientist, recognize as a description of my field.