I'll answer your questions in reverse order, if you don't mind, because the second one is more important.

> Is it really impossible to be 'good' if you don't follow God, according to scripture? Is an atheist sinful by definition?

It is completely possible to be good and not a Christian. It is completely possible to be a Christian that is a worse person than an atheist. There is actually a lot of confusion around this point, even among Christians. A lot of the problem centers around what "Christian" means. It could mean:

1) Someone who feels like they're a Christian. Even Richard Dawkins considers himself a "secular Christian" (a). He doesn't believe he's going to heaven or doing God's will on Earth.

2) Someone who Christ will admit into heaven when they die.

3) Someone who strives to follow Christ's teachings on this Earth.

For the purposes of this conversation, it sounds like you're asking about #2 (feel free to clarify if I'm making the wrong assumption here).

As far as that goes, gjm11 gets some elements right but seems to miss the mark for some reason. The Bible is very clear on this point. Salvation is given to people by God, through Christ. Period (b). We are supposed to have faith that He will be fair about granting eternal life. If you don't believe God is fair or that the Bible is true, then I suppose what the Bible teaches on this point is not interesting.

So what does God want from us? To trust Him, believe in Him, and have faith that He will take care of everything (c). It's hard to see how an atheist can have this perspective. It seems impossible to me.

How does this relate to being a "good person"? Well, eternal life is supposed to be the default state of things, but when we reject God and His teachings, we accept responsibility for our own eternity (d), and I certainly can't make myself live forever. When we are "bad people", it should be extremely obvious that we are rejecting God (e).

So while some of us may be good, or even great, people, all of us have rejected God on some level and blown our chances to enjoy entrance into heaven on our own merits (f). Christians who think they're "better people" than atheists need to re-examine the Bible and work on that part of themselves (g1) (g2).

On to the next question:

> ...but it seems strange to me that early Christians would be required to submit to a government that is persecuting those that attempt to obey God, for attempting to obey God.

It is strange from many perspectives. The Bible actually says that (h).

Anyway, Daniel (i) is a great book to read for an example of how to both obey God and submit to oppressive rulers. At times, Daniel and his friends would ask politely for exceptions from rules that would cause them to disobey God's commands (j). When that was not possible, Daniel and his friends would respectfully refuse and accept the consequences, even if that meant death (k) (l). That seems pretty harsh, but if you have faith that the all-powerful God is good and that He's watching you, all sorts of things are possible, even a peaceful death under persecution (m).

In certain ways, the physical world is the Matrix to someone with strong faith in God. When you have guaranteed salvation and are following the commands of an all-powerful God, you don't have to dodge bullets. You don't have to.

I realize that sounds silly to many, but it's what the Bible teaches.

(a) http://www.telegraph.co.uk/culture/hay-festival/10853648/Ric...

(b) https://www.biblegateway.com/passage/?search=john+14:6

(c) https://www.biblegateway.com/passage/?search=John+3%3A16&ver...

(d) https://www.biblegateway.com/passage/?search=Romans+6%3A23&v...

(e) https://www.biblegateway.com/passage/?search=Luke%2010:27

(f) https://www.biblegateway.com/passage/?search=Mark+10%3A17-27...

(g1) https://www.biblegateway.com/passage/?search=1+Timothy+1%3A1...

(g2) https://www.biblegateway.com/passage/?search=Ephesians+2%3A8...

(h) https://www.biblegateway.com/passage/?search=Proverbs+14%3A1...

(i) https://www.biblegateway.com/passage/?search=Daniel%201

(j) https://www.biblegateway.com/passage/?search=Daniel+1%3A8-21...

(k) https://www.biblegateway.com/passage/?search=Daniel%203&vers...

(l) https://www.biblegateway.com/passage/?search=Daniel%206&vers...

(m) https://www.biblegateway.com/passage/?search=Acts+7%3A54-60&...