The 1601 is EOL (end of life) but still definitely (note that it is spelled correctly) usable. As far as fitting into your network, it would replace your current router that terminates your T1. NOTE: you may want to compare your existing router to this one, as the 1601 is no longer supported by Cisco.
Connectivity...no need to convert DB60...that's crazy talk
The DB60 connector is your WAN interface (T1) (the connector under the blue line). Your Ethernet is under the yellow line and is inthe form of AUI and 10Base-T (RJ-45). Just use the RJ-45 port (unless you feel like hanging out with the other geeks circa 1985 and choose the AUI port)
config...I've only configured the larger Cisco's via the CLI...This 1600 has some sort of 'easy start' windoze-based config system...so it shouldn't be too hard...but if you have questions concerning syntax, let me know and I should be able to help you thru it.
Here is a link on Cisco with some information on the 1601