If you do a HEAD request, the server will most likely return the file size and the file's last-modified time. You can also do a conditional GET, by providing a "If-Modified-Since" header. In both cases, see the spec for details.

$ GET -m HEAD 'http://www.adaelis.com/misc/temp/monks.jpg'
200 OK
Connection: close
Date: Thu, 30 Sep 2004 00:04:47 GMT
Accept-Ranges: bytes
ETag: "3cb18d-f292-411ee589"
Server: Apache/1.3.31 (Unix) mod_fastcgi/2.4.2
Content-Length: 62098                          <-------
Content-Type: image/jpeg
Last-Modified: Sun, 15 Aug 2004 04:24:41 GMT   <-------
Client-Date: Thu, 30 Sep 2004 00:04:47 GMT
Client-Peer: 63.111.28.139:80
Client-Response-Num: 1
[download]

As for your second question, the minimum size required to get some basic info, why don't you take a peek into Image::Info's docs and guts?