will formatting local disk C lead to lose of internet connection ?

[Dunbar Pappy®ϟϟ]

Loss of some functionality is possible.
For instance, Vista/Win7 may be set to 'auto update' and drivers get mod's which would not be in the 'stock' disk used to format. I've seen cases just as so.
XP is different updating drivers, but may also have the problem.

For a less hassle re-install, I suggest having a CD with your current drivers.
With separate CD driver storage, accessing and installing functional drivers is far easier.
Try Driver Magician Lite (freeware);
http://www.snapfiles.com/Freeware/system/fwbackup.html
Use it to identify & back up all drivers for this BU. Put the file it creates onto a dedicated CD. Always keep it with your unit.
Update it periodically.