I don't believe Christians have a party, since there are pros and cons to all parties in the system. I know I generally vote Democrat since they tend to be more concerned about social services and aid.
Yes Christians can run, be involved, and especially vote. Since we are here, we should reasonably lobby for policies that reflect God's desire in society.
As far as a pastor serving in politics, that would depend on the position and the time requirements, since a pastor's primary focus should be on the ministry. Any time conflict should default to ministry works, not political duties.
Whether I would be affected by learning if the politician is a Christian, it appears most are as far as it's been discussed openly. I'd be concerned to how much their life reflects their faith.
I minimize my political involvement as the system is representative of the whole populace, and to avoid wasting precious time bartering with those with differing views, leaving that to those who are elected.
I don't see that our method of governing currently is even supported at all biblically, since there is no sovereign providence of God to account for. Having that be done in this age seems about as probable as cows growing wings and playing banjos.