I've been dealing with twisted pair cable since the mid-90s and I've never seen Cat 4 cable anywhere, commercial or residential.
Cat 5/5e both support GigE. The primary difference between FE and GE with respect to cabling is that GE uses all 4 pairs where FE used only 2 pairs. The difference between 5 and 5e cable is pretty negligible, and the GigE standard only requires Cat 5, not 5e
With respect to Cat 4, you're confusing the signal bandwidth and data rate. Cat 4 supports up to 20 Mhz signal bandwidth. It can be utilized by either 10BASE-T (10 Mbit/s) or 100BASE-T4 (100 Mbit/s).
If there's twisted pair data cabling in the home at all, it's probably suitable for GigE. Otherwise it's likely RJ11 phone cabling that's not typically going to be in a home run topology.
That said, the standards are pretty forgiving over shorter distances. Here's someone claiming they got GigE speeds over Cat 3 cabling in an older home: