Is it possible to do an in-place upgrade of SQL Server 2005 Standard to SQL 2012 Enterprise?
Yes, according to Supported Version and Edition Upgrades you can upgrade in place from 2005 Standard to 2012 Enterprise, provided SQL Server 2005 is currently on an operating system supported by SQL Server 2012 Enterprise (meaning forget about XP or 2003 RTM) and that the architecture is the same (e.g. 64-bit -> 64-bit).

Though you should do step 1 and 2 anyway so you have something to compare to and fall back on, since you never know what might go wrong during the upgrade. (And don't forget to knock on wood and all that.)

Personally I prefer a side-by-side installation onto a new machine, especially if the 2005 hardware and/or operating system is old. The performance benefits you get "for free" when you upgrade are amplified immensely when you also upgrade the underlying stuff as part of the process.