perl -al0nF: -e 'print /name/&&++$x,/name|date/&&$F[1],/date/&&$/' data.txt

##</code><code>##

name: url1.domain1.com
date: 2004/2/1
unwanted info: blah blah blah

name: url2.domain5.org
date: 2004/3/2
unwanted info: blah blah blah

name: url3.domain1.com
date: 2004/2/3
unwanted info: blah blah blah

name: url4.domain5.org
date: 2004/3/4
unwanted info: blah blah blah

name: url5.domain1.com
date: 2004/2/5
unwanted info: blah blah blah

name: url6.domain5.org
date: 2004/3/6
unwanted info: blah blah blah

##</code><code>##

1 url1.domain1.com 2004/2/1
2 url2.domain5.org 2004/3/2
3 url3.domain1.com 2004/2/3
4 url4.domain5.org 2004/3/4
5 url5.domain1.com 2004/2/5
6 url6.domain5.org 2004/3/6