When you add her income, you get more taxable income. She may have had no taxes withheld (look at box 2 of her W2) or she may have had very little withheld. When you combine both incomes, her income would be taxed at your rate instead of the rate for someone making less than $12,550. If she is still working then she needs to change her W4 so that they would withhold taxes at the rate of your incomes combined instead of at her lower rate.
If you are married and living together, you cannot file as head of household. Your only options are married filing jointly and married filing separate. You cannot claim a spouse as a dependent.
You can claim your 3 kids on your return and she can file her own return without claiming the kids but you would be filing as married filing separate not head of household.
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