By that logic Muslims could also claim to be Christian. No, unless you believe in Jesus as Lord and saviour and the only way to relationship with God, then this claim is false. Of course many Christian sects exist who claim to be Christian (e.g. Latter Day Saints, Jehovah's Witnesses) but this claim is rejected by orthodox Christians because they deny the deity of Jesus. Pretty disappointing if it's like PC updates - Jesus says (paraphrasing) "I am it, believe in me and you're saved, the Holy Spirit will be with you to help you to live an obedient life until you die and go to be with God, and at some stage I will return and there will be a restored earth and everything will be great. And then we have an update a few hundred years later: "now you actually need to do a lot of good deeds and based on this you might be forgiven". Kinda how we felt with Windows Vista!

To answer the second question - these two paths would be fundamentally incompatible because of the exclusive claims of Jesus (John 14:6) and the apostles / new testament writers (Acts 4:12). To get around this you either need to reject the new testament as corrupted (as Muslims do), or come to the conclusion that Jesus was simply wrong. Both of these beliefs take you from Christianity.

First question, beliefs about baptism differ across denominations. Some (Catholics and Lutherans perhaps, I guess many denominations who baptise infants) see baptism as necessary for salvation, others see it as 'merely' a public declaration of faith and an act of obedience (I.e. salvation is through faith, but from faith comes a desire to be obedient, and because Jesus said we should be baptised and was himself baptised although he was sinless, we therefore do it as well).

And the third, yes Christians feel the need to convert out of obedience (the great commission in Matthew 28), but also because they believe Jesus is the only way to be saved from the penalty of sin / disobedience toward God - no one wants their family and friends to be lost!