Well that's a load of nonsense - defies all known electrical theory - if voltage is lower, then current into whatever load is presented is also lower.
So Fuse blowing because of low voltage is absolute rubbish
Yes it is logical that the starter would not run because the headlight fuse was blown - question is why did it blow and it was NOT because of low voltage

The battery not being optimal is incidental - honestly doubt if it was related at all and possibly still serviceable (but without any first hand observation I won't argue that point)
Don't understand the last point - fuse was not in the fuse box???