0. E7 supports two-way, four-way or eight-way, while E3 supports one-way only.
1. Better check with board manufacturer.
3. If you don't need more than one CPUs in your system, choose E3. E7 family is for high end server systems and mission-critical applications.
The E3 is a entry level server / workstation product where as the E7 is a top end server processor usually utilized in multiple CPU configurations.
The E3 uses the LGA 1155 socket and the E7 uses the LGA 1567 socket.
The E3 v2 has been seen to work in a number of desktop boards released for the Ivy Bridge range of processors but I believe this is not officially endorsed by Intel. ASRock and ASUS (being the same company really) both list E3 v2 CPUs on their Z77 (possibly H77) boards compatible processor lists.
I do not believe that Intel makes a standard desktop format E7 board which is why Edward is unable to recommend one for you. Remember the E7 is not intended to be used for a desktop processor.
The E7 is a superior processor but just like buying a Ferrari Engine, you need to spend big money on the rest of the car to make it go.
My suggestion is to get an E3 v2 or some sort and, as Edward has suggested, an S1200BTLR (ATX), S1200BTSR (MATX) or S1200KPR (mITX). There are other boards out there by other manufacturers so have a search for C202, C204 or C206 chipset motherboards.