Is Orthodoxy more important than Orthopraxy?

1 Corinthians 11:13 and 11:16 indicate that this was an issue for the Corinthians specifically. "Judge in yourselves: is it comely that a woman pray unto God uncovered?" Most Americans would say "Huh?" While the Corinthians would say, "Well, no." 11:16 says "We have no such custom, neither the churches of God". In other words, this issue was specific to that culture.

1 Peter 1:3 is seen as an admonition to decorate the inside, and make that the top priority, not a prohibition against decorating the outside. I would say that Biblical literalists would enjoin the admonition of modesty, which culturally changes from venue to venue. 1 Corinthians 6:12 states that "all things are lawful unto me, but all things are not expedient." So these are seen not as specific prohibitions to be followed but as specific applications of an overall rule of modesty and prioritizing the inward person over the outward appearance.

If you take Peter straight, then your females have to be prohibited from "putting on of apparel"; they will have to go naked. :) Thankfully, 1 Timothy 2:9 clarifies this when Paul says that women "adorn themselves in modest apparel". So the rule is modesty, and the examples given were not modest in that time and culture.

Tell a woman to dress modestly, and she will know what that means in her culture. What is immodest in American culture? Dresses too high, too low-cut, too suggestive, too gaudy. Excessive jewelry, not jewelry altogether. Makeup in America is irrelevant to modesty in most cases. But she already knows this.

Conclusion: If you tell her to dress modestly, you are helping her focus on the inward person. If you tell her she can't wear this or that, or she has to wear this or that, then you fall into the same trap of focusing on the OUTWARD person when the Bible passages you cite are trying to get your women to focus on the INWARD person.

That is why Biblical literalists don't make rules governing what women can and cannot do on the outside. It would negate the inward focus of the passages in question.

Short Answer

Hebrews 11:6 says this:

And without faith it is impossible to please God, because anyone who comes to him must believe that he exists and that he rewards those who earnestly seek him.

It does not say this

..must believe that he exists and that he rewards those who do the right things.

Likewise, John 14: 15 says this:

15 “If you love me, keep my commands.

not

If you keep my commands, I will love you.

1. Orthodoxy over Orthopraxy is distinctively Christian

I would argue the primary distinctive (beyond the obvious claims that Jesus rose from the dead) of Christianity above and beyond just about every other religion is that Christians are more concerned about Orthodoxy (right belief) than Orthopraxy (right action).

As a simple exercise, go look at Islam.SE and Judaism.SE. There, most of the questions are about "How do I correctly do X?" Is it halal? Is it haram? These are the common fodder. And, in a works-based religion, this would be the focus.

2. Grace is central to what we believe

In contrast, Christianity has been typified by its concern for Orthodoxy from the beginning. Having realized that even "our righteousness is as filthy rags" (Isa 64) in the sight of a God whose righteousness is not even comprehendible by man (Isa 55), Christians understood early on that it was only God who could "make us stand in His presence, blameless with great joy" (Jude 24).

When Paul makes grace the central focus of the Gospel (Eph 2:8-9), when Jesus says "apart from me you can do nothing" (John 15:5), and when he says that you cannot enter into the Kingdom "unless your righteousness exceeds that of the Pharisees" (Matt 5:20), it leads many to suspect that God ultimately desires "mercy and not sacrifice". That "sacrifice of obedience" is a free will act of violition - not action. As such, grace truly is, as the song says, "greater than all our sins".

3. Jesus hated Pharisees

The opposite of this tack truly is, as others have said, the central issue that Jesus had with the Pharisees. They would "tithe mint and cumin" (Mat 23:23) and neglect what Jesus himself declares "the more important matters of the law--justice, mercy and faithfulness. You should have practiced the latter, without neglecting the former." Jesus hated these people. He called them hypocrites and vipers, and said of them "You travel over land and sea to win a single convert, and when you have succeeded, you make them twice as much a child of hell as you are!"

All of this is fine Sunday School fodder, but what most people forget is this - Pharisees were good people. They really did observe the law. They really did help the widow and the orphan. They really tried to be good. But they couldn't. They were fallen human creatures in need of grace.

4. Conclusion: If you want Orthopraxy, Go be a Pharisee, or a Muslim. But if you want the Good News, Be a Christian

I'll say it plain - If you care more about what a person does than what a person believes, then you should be a Muslim. They are far more observant than most Christians.

But we have a Savior who declares us good. We are not only not good at doing good, we are dead in our ability to do good. As Paul wrote in Romans 7:

Once I was alive apart from the law; but when the commandment came, sin sprang to life and I died. 10 I found that the very commandment that was intended to bring life actually brought death. 11 For sin, seizing the opportunity afforded by the commandment, deceived me, and through the commandment put me to death.

The very Good News of Scripture is this:

While we were yet sinners, God died for us.

That I (and all Christians who believe in Grace) believe is the central tenant of the Bible. God who is all powerful saves. I, as his fallen Creature cannot. If I believe this, then what can I do to make God love me more? What can I do to make him love me less? Nothing. As such, what I do is irrelevant. What He does is everything.