Oh, very interesting. I find Arthur a very interesting character, because the glimpses we get of him, the way other people see him, draw a very heterogenic picture. I give this a shot! What follows is purely my interpretation.

1: What are his powers?
Whetever his powers are, I think it was something very powerful and potentially dangerous. It could even be that his suicide was to prevent his son from absorbing it. I can see that he did something terrible with it in the past and stopped using it altogether. For me, it's something that doesn't fit in with everyday life as well as others (e. g. mind-reading or prophetic dreams, which can be used on a regular basis without anybody else knowing).
2: Is he dead? If so, how and why?
Yes, I think so. See above.
4: What sort of personality does he have?
I see him as a very strong, quiet character who can nevertheless have quite a temper if something strikes a nerve. The way he and Nathan got along makes me think he was similar in character to his eldest son. I think he was a very traditional, conservative man, too.
5: What should his horoscope be?
No idea?
6: how would he react to Angela's affair?
I see him as the type of man that would hardly react at all - because he would have been deeply hurt and wouldn't have had a way to react to it adequately. It probably hasn't been mentioned since right after it came out.
7: Him as a Father?
Very different for Nathan and Peter, I think. They ar so far apart in age
and temper that they probably knew a very different Arthur. Nathan clearly seems to have taken his father as his role model; I think they got along very well. Arthur would have been proud and taken this sort of hero worship for granted, so he woulnd't have taken it well when it became apparent that Peter didn't share Nathan's adoration of him. (Extra twist would be that Peter seems to have hero-worshipped Nathan in much the same way that Arthur felt would have been
his due. The problem probably augmented itself. Peter felt Nathan was more present as a father figure when he was little than Arthur was; this angered Arthur, so became even less present. There seems to have been a severe breach when Peter decided to become a nurse. I don't think it was as dramatic as some fanfic writers make it - it does seem very demonstrative that Arthur didn't come to Peter's graduation party, but I don't think that there was complete silence between them (or Peter wouldn't have been so disappointed, which he was).
8: His complex relationship with Lindeman explained.
Can't really comment on that. It seems that theirs was never a fully easy friedship - Linderman saw Arthur as weak, Arthur might have felt indebted to Linderman, but not really as best buddies.
9: Adam and him?
Can't really comment on that either. Adam seemed to have them all pretty well in hand.
10: How did he deal and feel about Claire?
I could imagine that he would have been content to have Angela deal with the details. He would have been very disappointed in Nathan - similar to how he felt when Angela betrayed him.
11: What are the Petrelli family rules?
Fit in. Do what's expected of you. And then some, in a good, patriotic way.
12: An overview of how he feels about Angela.
A lot of guesswork there, she never talks about him as much as her sons so. I can see their relationship as having become a bit stale, perhaps. Probably with the realisation that they weren't going to make the world a better place.
13: His status within the company.
I think he would have been content to
feel important. I always thought Angela was more active there.
14: What actor should play him?
No idea.
15: Villain or Hero?
Much like Angela - they started out wanting to save the world, and became disillusioned. I think he wouldn't have been as much of a schemer, but I can see him pretty much retreating from active duty in that department in later years.
16: Overview on his crazy?
His earlier depressions were probably made worse by the things he saw and the realisation that he wasn't going to make a difference.