Income tax information?

You can file with a status of Married Filing Separately, but you can also file Married Filing Joint. The downside of MFS is that you pay a higher rate and are ineligible for a number of tax credits. The upside is that your wife's income does not count assuming she is what the IRS calls a "non-resident alien". If you decide to file Joint her income counts, regardless of where it was earned.

If you have a child that lives with you it is possible you could file as Head of Household which is a better deal than Married Filing Separately. You should check with a professional preparer or visit a VITA site to understand your options.

To file MFS when your spouse doesn't have an SSN or ITIN, see the instructions for the 1040. You will write NRA in the SSN field and mail the return in.

You can elect to file MFJ, see IRS pub 519.

If your spouse lived with you after June and is a resident alien, you cannot file HOH. If she was a non resident at any time during the year, HOH may be possible even if she lived with you, but not for EIC.

Note, an undocumented alien (here illegally) is not the same thing as a non resident. Living in the USA all year would meet the substantial presence test for tax purposes.

See IRS pub 519 for this HOH exception.

Yes. Everyone who is married can file as "married filing separately".

she had to have either a SSN or ITIN to be reported on your tax return

since she does not live in this county(any chance she lives in Mexico or Canada) most programs in use will not allow the married separately status to enter if you do not have the proper ID number

What country? Is your wife a resident alien or not? Do you have children? All this matters.

Plain old Jim's first paragraph is correct, but no you can't file as head of household.