I also have the same issue as the OP with the Broadcom Ethernet driver dropping my internet connection. I've also tried everything the OP did (including manually installing newer driver V, but ALL to no avail.

It got to a point where I ended up buying Plugable's USB 3.0 Gigabit Ethernet Adapter: http://plugable.com/products/usb3-e1000 and this is also dropping the Win 10 built in ASIX drivers where I have to continuously reinstall the drivers/Rollback/unplug the USB 3.0 adapter etc.......all to no avail.
Plugable support is quite good/actively communicate with me and are sending a free replacement (in case this one might be defective), but I'm a bit skeptical and think it's more of a Win 10 driver issue.....IDK.

Anyone know of a solution to this?
I'm really trying to avoid reverting back to Win 8.1 as I do like Win 10 and EVERYTHING else is working perfectly.........just not the internet.......