Why did the Vikings only set up trading cities on the coast? Why did they never invade, conquer and rule the other Kingdoms?

It's a lot easier to raid small, relatively defenceless settlements one at a time, and build your own in previously sparsely populated areas (many of these coastal locations were uninhabited prior to viking settlement), than go about invading existing kingdoms.

Attempts were made to do so, but when going up against Irish kings (in either an offensive of defensive capacity) the vikings suffered their fair share of defeats, before being effectively, ultimately defeated at the battle of clontarf in 1014.

https://en.m.wikipedia.org/wiki/History_of_Ireland_(800–1169)

This same source goes a long way to answering your other questions;

q3),q5) re genetic influence:

This Norse influence is reflected in the Norse-derived names of many contemporary Irish kings (e.g. Magnus, Lochlann or Sitric), and DNA evidence in some residents of these coastal cities to this day. A genetics paper in 2006 by Dr Brian McEvoy found that most men with Irish-Viking surnames carried typically Irish genes

q4),q2) re cultural and religious influence:

Inter-marriage between vikings and Irish is well documented, including mention of vikings converting to christianity upon marriage. Viking technology and artistic influence was adopted in Ireland, yet customs-wise the vikings assimilated into Irish culture, rather than vice versa. Later raids saw vikings begin to spare churches, and viking coins from Dublin have also been found to be printed with the cross symbol.

https://dh.tcd.ie/clontarf/Religion%3A%20Heathens,%20Christians%20or%20Something%20in%20Between%3F