If "obscenity, foolish talk or coarse joking" are immoral (which I agree with), then how do we as Christians, who encounter these jokes every day, continue to live a pure life, staying away from these things? How do we continue to be kind to those who do such things and steer far away from the stereotype that Christians think that funny is equal to sin? If someone tells an off color joke and we do not laugh, or ask him not say those kind of jokes, it could cause a block between you and that person. In my life this is my girlfriend. Although she isn't totally 'off the deep end' with these jokes, she still says them, laughs at them, and then says she doesn't care. She is a Christian.
How far does "righteous anger" go, because I feel angry a lot of the time due to my own sins and the acceptance of sins among others?
What makes something right or wrong has nothing to do with your agreement. Inappropriate language is wrong because God said so. "But fornication and all uncleanness or covetousness, let it not even be named among you, as is fitting for saints; neither filthiness, nor foolish talking, nor coarse jesting, which are not fitting, but rather giving of thanks. For this you know, that no fornicator, unclean person, nor covetous man, who is an idolater, has any inheritance in the kingdom of Christ and God. Let no one deceive you with empty words, for because of these things the wrath of God comes upon the sons of disobedience" (Ephesians 5:3-6).
Similarly, just because a person claims to be a Christian doesn't mean they are actually following Christ. "But why do you call Me 'Lord, Lord,' and do not do the things which I say?" (Luke 6:46).
When people are involved in sin, Christians don't follow along. Period. It doesn't matter if others give you funny looks or not. "Therefore, since Christ suffered for us in the flesh, arm yourselves also with the same mind, for he who has suffered in the flesh has ceased from sin, that he no longer should live the rest of his time in the flesh for the lusts of men, but for the will of God. For we have spent enough of our past lifetime in doing the will of the Gentiles -- when we walked in lewdness, lusts, drunkenness, revelries, drinking parties, and abominable idolatries. In regard to these, they think it strange that you do not run with them in the same flood of dissipation, speaking evil of you. They will give an account to Him who is ready to judge the living and the dead" (I Peter 4:1-5). Being a Christian is not about being accepted by the world. Our job is to stand out from the world. "Do all things without complaining and disputing, that you may become blameless and harmless, children of God without fault in the midst of a crooked and perverse generation, among whom you shine as lights in the world, holding fast the word of life, so that I may rejoice in the day of Christ that I have not run in vain or labored in vain" (Philippians 2:14-16).
If your girlfriend is attracted to sin and doesn't care that what she does is wrong, then it is past time you found a better woman because this woman will bring you down, not build you up.
Sin is going to being in the world, so long as this world continues to exist. "I wrote to you in my epistle not to keep company with sexually immoral people. Yet I certainly did not mean with the sexually immoral people of this world, or with the covetous, or extortioners, or idolaters, since then you would need to go out of the world. But now I have written to you not to keep company with anyone named a brother, who is sexually immoral, or covetous, or an idolater, or a reviler, or a drunkard, or an extortioner--not even to eat with such a person" (I Corinthians 5:9-11). As Christians we understand that worldly people are going to behave in a worldly manner. We don't go along with it, but we can't pretend it doesn't exist. If we feel anything, it ought to be pity that they are so blinded by sin that they don't see where they are going.
It is a bit different when people claim to be religious and act anything but. These are the people who give Christianity a bad name. So yes, it is appropriate to get mad at yourself for getting caught by sin -- provided that you do something about it. "For observe this very thing, that you sorrowed in a godly manner: What diligence it produced in you, what clearing of yourselves, what indignation, what fear, what vehement desire, what zeal, what vindication! In all things you proved yourselves to be clear in this matter" (II Corinthians 7:11).
Getting mad at others also is not productive unless it drives you to help people change their behavior. "For we must all appear before the judgment seat of Christ, that each one may receive the things done in the body, according to what he has done, whether good or bad. Knowing, therefore, the terror of the Lord, we persuade men" (II Corinthians 5:10-11).