The private ranges defined by RFC 1918 are:
10.0.0.0 --- 10.255.255.255 (10/8 prefix)
172.16.0.0 --- 172.31.255.255 (172.16/12 prefix)
192.168.0.0 --- 192.168.255.255 (192.168/16 prefix)
The DHCP service included with most routers can be configured to allocate local IP numbers in any of these. I don't think anyone other than IANA can allocate IPs in the 5.*.*.* range.
Hamachi does provide a proxy so that you could establish a VPN to your wife's laptop but if that were all you wanted from it, there are other ways (e.g.
http://www.dns2go.com) but costs may not be negligible.
Hamachi would appear to have merit, but for a network of your size with one fixed IP base and two laptops, you ought to be able to achieve all you want using facilities already there. You say that you want to be able to connect from home to your wife's machine when she is out and about. That will be difficult if, as is likely, she is only intermittently connected and, even more likely, has dynamic IP. (Since only she knows when she is connected, it would make more sense in most cases to let her initiate the connection.)
One point in favour of hte Hamachi system is the use of Public Key Encryption which would be very helpful in setting up connections with larger numbers of unknown members. Encryption is easier (and possibly more secure) using symmetric keys which need to be entered into both machines - not difficult when they are in the same location at times,
http://www.draytek.co.uk/support/vpn_setup2.html provides information on how to set up a "Remote worker" VPN but it relates to a Draytek router so will need additional information for your own, especially if it does not include a VPN server.