Strange access requests

There is no reason for concern in the log entries you posted.

The first is a request for an image file that many browsers use as icon when displaying a page from your site or a bookmark.

The second request is an attempt to use your server as HTTP proxy. However it looks like your server just ignores the proxy part of that attempt and instead responds as if it had been a local site.

If you want to be certain, you can test it manually using telnet. Here is what it looks like on one of my servers:

$ telnet localhost 80
Trying 127.0.0.1...
Connected to localhost.
Escape character is '^]'.
GET http://s1.bdstatic.com/r/www/cache/static/home/img/logos/nuomi_ade5465d.png HTTP/1.1
Host: s1.bdstatic.com

HTTP/1.1 404 Not Found

And here is what it looks like if I try to access the root of the domain:

$ telnet localhost 80
Trying 127.0.0.1...
Connected to localhost.
Escape character is '^]'.
GET http://s1.bdstatic.com/ HTTP/1.1
Host: s1.bdstatic.com

HTTP/1.0 302 Moved Permanently

Both of the above were served by my default vhost.

In your case I would assume you have a default vhost configured to redirect everything to your actual domain.

Point of information -- we saw a few requests for this exact same URI (s1.bdstatic.com) come through one of our addresses a few days ago. So i'm reluctant to assume the request is benign, although it could just be someone using that image in a scan for open proxies.

125.64.35.67 1418849297 <internal> GET http://s1.bdstatic.com/r/www/cache/static/home/img/logos/nuomi_ade5465d.png HTTP/1.1
125.64.35.68 1418853519 <internal> GET http://s1.bdstatic.com/r/www/cache/static/home/img/logos/nuomi_ade5465d.png HTTP/1.1

It's a scan for open proxies. I've seen a few in the last couple of weeks.

125.64.35.68 - - [05/Jan/2015:03:54:36 -0500] "GET http://s1.bdstatic.com/r/www/cache/static/home/img/logos/nuomi_ade5465d.png HTTP/1.1" 403 254 "-" "Mozilla/4.0 (compatible; MSIE 7.0; Windows NT 6.3; Trident/7.0; .NET4.0E; .NET4.0C; .NET CLR 3.5.3072; .NET CLR 2.0.50727; .NET CLR 3.0.30729; Tablet PC 2.0)"