I only earned $3000 independantly, and I used unofficial support my husband gave me to pay all the bills for our household. In other words, I physically budgeted that support and paid the bills with it. Can I file Head of Household since my children lived with me the entire time, never staying with him, even though I didn't independantly earn the money to pay the bills?
One of the requirements for filing as head of household is that "you paid more than half the cost of keeping up your home for the year." Once your husband gives you the "unofficial support" it's your money. If you used it to pay for keeping up your home, then you paid more than half the cost of keeping up your home. So yes, you can file as head of household. (You appear to meet all the other requirements.)
Filing a tax return with only $3000 of earned income will get you a very small amount of earned income credit --- about $1200 --- and maybe a few dollars of additional child tax credit. You only get additional child tax credit for the amount over $2500----so your ACTC would be less than $40 for $3000 of income from working, assuming all the kids are under 17.
The child support you received is not taxable and is not entered anywhere on your tax return. The child support is not used to calculate any child-related credits you can receive. Only income earned from working is used to calculate child-related credits.
If you are still legally married you do have the option of filing married filing jointly with your spouse, even if you do not live together.