There's a hidden file with extension .padl in my home directory and I don't know what it's for

First, go to Document --> Settings --> Modules, add the Customisable lists (enumitem) module:

For LyX 2.3+

After adding the module, you can use the Enumerate-Resume layout. This will resume previous enumerations, if they exist, both in LyX and the output document.

Alternatively

Write the two lists with text between them, and to add the [resume] option, right click the first entry in the second list, and choose Enumerate options (shortcut Alt + A1):

In the inset that shows up, type resume:

LyX and output:

This is actually a known feature of Windows XP/2003. I'm not sure if Windows 7/2008/R2 retained this behavior.

The working set of an application is trimmed when its top-level window is minimized
http://support.microsoft.com/kb/293215

Yes, all home gateway products allow you to manually configure a static public IP address on the Ethernet WAN port.

Every router I've ever come across lets you do this.