If the property was your homestead at the time of the foreclosure, you likely qualify under the Forgive act. As for your cars, it depends on the equity in the vehicles (balance owed minus value). If you are very upside down, you may consider a redemption loan on the vehicles.
If you received a discharge in the Chapter 7, you can't receive a discharge in a Chapter 13 case which is filed within 4 years from the date you filed your Chapter 7. You may be able to file a Chapter 13 to get more time to pay the tax debt but you won't be eligible for another discharge. I am not sure what you mean by not including your house in the bankruptcy. If you listed it and the mortgage debt, even if you claimed the house as exempt, you may have discharged any personal obligation for the deficiency in the Chapter 7. You might want to check with your bankruptcy attorney on that.
If you received a Chapter 7 discharge from your 2010 bankruptcy filing and did not reaffirm the mortgage, than the mortgage debt would have been discharged in the bankruptcy (although the lien would still remain). The discharge would protect you from a future deficiency judgment.
For tax debt to be dischargeable in bankruptcy it must satisfy three criteria: (1) it must be at least 3 years from when the taxes were due; (2) it must be at least 2 years from when the returns were filed; and (3) the taxes could not have been assessed in the past 240 days.
In addition to whether your tax debt is actually dischargeable, you must satisfy the time period between successive bankruptcy filings in order to be eligible for another discharge. You did not say what chapter you filed under in 2010. For a chapter 7 to chapter 13 filing, to get a discharge you need to wait 4 years between filing dates. For a chapter 13 to chapter 13, you must wait 2 years between filing dates to get a second discharge.
Please note that there is no such thing as "not put the home in the bankruptcy ."
*You* go into the bankruptcy, and all of your debts and assets follow you.
It sounds like it was deliberately not listed, which could exempt it from your prior discharge.
Even though you're not eligible for a discharge, you could conceivably file a 13 now and make your payments, and then dismiss it in 2014 and file a new one, in which you could get a discharge. You need to discuss with an attorney in your district whether or not this would be considered good faith; the answer varies throughout the country.