“Do we need to fill out a schedule E to report the income?”=======>In genral yes; if you rent it for fewer than 15 days, then you do not need to report rental income on your return.So, you need to report your rental income/exp on Sch E of 1040. If you have any personal use of a vacation home or other dwelling unit that you rent out, you must divide your expenses between rental use and personal use. If your expenses for rental use are more than your rental income, you may not be able to deduct all of the rental expenses
“We earned roughly 10K through this rental for 2012. Since this was our only income for 2012 do we still qualify for Earned Income Credit?”===>I don’t think so; it is not being considered "earned income" for EIC or IRA purposes. Rental Real Estate Income is unearned incomeUNLESS you are r/e pro.
“ Or is this type of rental considered investment income and we won't be allowed to claim the credit because it is over the $3,200 limit?”===>as said above.