Please keep in mind that I'm a libertarian, not a Republican, but perhaps I can help you see an alternate opinion?

So republicans like not having a big evil controlling government--but doesn't that set the path for big evil controlling companies to abuse lack of laws?

Here's the truth: Evil controlling corporations pay politicians to legislate in ways that help them. These corporations are able to get as big as they are because of the laws. In a truly free market, there would be way too much competition for any one company to take over. Right now, both Republicans and Democrats practice crony capitalism and it's ruining the country.

If republicans are in support of small businesses, wouldn't more government regulation actually support that,

Same as above. There are lots of policies that weed the little guys out.

And how is it so evil to pay taxes for necessary services like education?

I think their thought process is that we should streamline the system. Make it leaner and more cost-efficient.

There seems to be this mentality of "individual responsibility"--but that overlooks how necessary it is to actually have a functioning government that provides security for its citizens.

Republicans have no problem with taxes for fire departments, police, military, etc.

except to stop others from being entitled to the same individual freedoms everyone should have (marriage, some instances of immigration, a woman's right to her own body).

True, Republicans do this. So do Democrats. Democrats have traditionally supported the drug war, banning big sodas like in NYC, banning trans-fats, restricting gun rights, etc. Democrats might be for gay marriage, but what about other types of marriage like polyamory? I've only seen libertarians supporting everyone's rights across the board.

How can lawmakers actually say mindblowingly stupid shit like, "a woman's body has defense mechanisms against rape", but people like that still have political power?

Because normal people don't choose the politicians anymore. Corporations do.

They don't want big governments, but big fat military spending is A-Ok.

Agreed.

This last part must be a polarizing issue for republicans--I remember vaguely as a teen during the Bush admin, a lot of republicans were vocally in support of the freedom act to prevent terrorism; but beefed national security doesn't necessarily go hand-in-hand with 'evil big brother is listening'--so where do a lot of republicans lie on that matter?

It is a polarizing issue for them, you are correct. The more libertarian-leaning Republicans are against it. The old-school Republicans are for it. Similarly, the old-school Democrats are for Obama's expansion of the NSA. Others are not.