1. Can I immediately move my Dudes from home to an Out-of-Town deed or do they have to be in a town square area first?
A dude can move anywhere by booting.
2. Does Tommy Harden's trait triggered in the Gamblin' phase when the opponent reveal a cheatin' hand?
The card says: "in Tommy's shootout". Lowball is not a shootout.
3. Does 1 Control Point gained by the Cattle Market stacked when there are more than one other ranch in play? I think not.
It's a yes/no condition. Is there another ranch in play? If yes, gain 1 control point.
4. Does Cattle Market gained 1 Control Point when my opponents play a Ranch of their own?
Yes. The card does not specify that the ranch must be owned or controlled by you.
5. Can I have more than one of the same Deed in play? As far as I know, Deeds and Goods are not considered unique.
Rulebook p.14: "All dudes and deeds in Doomtown are unique".
6. When a player have Bank of California in play, will the opponents be able to take the Ghost Rock in which you have ante during Gamblin' phase and if they win in the lowball poker?
Yes. So far the only card that is affected by the Bank is This is a Holdup!
7. Just to confirm, Pearly's Place enables a player to play two shootout plays if they have the initiative after winning at the Gamblin' phase, right? If a player have 2 Pearly's Place, they can have 3 shootout plays consecutively.
Yes to first, No to second. After one Pearly's Palace effect is used, the condition "before any player makes a Shootout play" is no longer met, so the other Pearly's cannot be used. You can make a normal shootout play, though, if you won lowball.
8. Yan Li's Tailoring will allow a player to raise the influence of a chosen Dude by 1. Will this increase be permanent? I should think so, because from p. 26 of rule, it says, "After effects places a counter, it's the counter that has ongoing effect. In effect, counters are permanent changes."
Yan Li's Tailoring does not place any counters. If you mark this temporary increase by placing a chip on a card, that doesn't mean it's a counter in the sence that you refer to
Yan Li's influnce increase follows the rule given on p.26 of the rulebook: "Effects caused by a Noon ability last until the end of the day".