How can a Christian reject part of the Bible and still call themselves a Christian?

[Sword of Christ]

Oh, I get it, you're one of those namby-pamby Christians who thinks that somehow human culture overrides The Word of God.

If we override Gods Moral Law then that's a sin or potentially outright evil. You don't need a book to tell what right and wrong is that should be internal within you. The Bible just points out to people who believe otherwise that our morality is sourced from God and isn't just an opinion of our own minds. Moral values exist regardless of our own opinion either way, there is a right and a wrong and a good and a evil.

Your kind tries to accommodate "modernity"

There's nothing particularly modern about this at all it's a fairly ancient understanding of the Bible if anything. Certainly as old as Christianity itself.

and say things like "well, you know that Yahweh was really against slavery all along but human beings in that cultural context and time weren't ready to abolish slavery and so..."

We would probably still have (legal) slavery today without our Christian values we were the ones to abolish it the rest of the world caught up somewhat later.

"From approximately 650 until around the 1960s, the Arab slave trade continued in one form or another."

as if Yahweh were powerless to command its abolition if that were indeed his will.

It did actually happen so there you go. You can't argue with the results.

is true with homosexuality.

You just have to apply the principals of human equality and rights to homosexuals as we apply them to everyone else. It's likely they put that in there because it was something the pagans around them were heavily into and there was a desire to distinguish themselves and remain culturally separate as a distinct people.

I repeat, if it IS The Word of God, it is timeless and universal, not relative to fickle human whims and relative cultural context.

The whole Word of God isn't contained in any visible book it's Christ/the Logos of God and within yourself. The Bible is directs you towards this truth you already know on some level, see it as a guidebook.

The Bible supports the divine right of kings (Rom 13:1-3). It admonishes slaves to obey their masters (Eph 6:5). Jesus rejected the idea that all humans are equal (see "Cannanite Dogs" story, Matt 15:22-28). Women are to submit to men and have no position of authority (talking to you Sarah Palin, Michelle Bachman, et al) (1Tim 2:11-12).

"And Jesus answering said unto them, Render to Caesar the things that are Caesar's, and to God the things that are God's. And they marvelled at him." Mark 12:17

That seems compatible enough with the secular\democratic state, you keep your religion and your government separate. In the Quran it has to be all one thing but Jesus never advocated it. You'll notice democracy and secularism didn't really take off in the Islamic countries. So you can see the influence a religion can have on the values of a culture. I'd argue Christianity has founded a pretty damn good one.

Christianity is only in its current unfortunate state because it had to compromise with secularism, science, democracy and modernity.

I'd argue that Christianity founded all of these things, or at least re-discovered and developed upon the science and government systems of the Classical world. It did take have wait until religious authoritarian Catholic state had lost much of it's power but that was something developed by the Roman Imperial state. So the society in which we now live is a more authentically Christian one.

The Golden Age of Christianity was the Dark Ages when science and secularism were kept in check.

The last 500 years has been the golden age of Christianity.