Sorry Mike, I refuse to leave the router on over night, it works perfectly satisfactory that way and has done ever since the advent of ADSL I connect with exactly the same speed as I had at close down around 7.6Mbs which on a nominal8Mbs connection is fine.
As regards your other issues, they are not related and are more likely to be due to a failing CMOS battery as mentioned above.
The question asked was
why a failing battery should cause it.
Bit-twiddler, thanks it could well be a clock related problem, but how would it see a public network before the router had initialised?