White mineral oil is basically what most fretboard conditioners are, except those have added stuff in them which may or may not be bad for the fretboard.
I know the dunlop lemon oil fretboard conditioner is mostly made up of white mineral oil + some other stuff. It isn't really lemon oil. Its a petroleum distillate, as is white mineral oil.
I'd rather use white mineral oil as it has no smell and no other added stuff ( solvents/cleaners) inside it.
I got a big bottle for RM6 at guardian plaza pelangi. You can get it from guardian SG for roughly 3 bucks.
I apply it with pieces of old cotton T shirts, cut into 3 inch squares.
Oiling a fretboard happens and should be done when the natural oil of rosewood evaporates, thus leaving the fretboard dry.
A dry fretboard is prone to cracking and you'd risk chipping it when having a re-fret. Oiling a fretboard basically replaces the natural oil lost in the board bringing it back to life.
Over time, the mineral oil gradually replaces the natural oils in the wood, you'd want pure mineral oil without solvents. If there are solvents, you run the risk of having them dissolve glue in the neck/frets.
A little oil goes a long way, I use 1 drop per 2-3 frets. I do it once or twice a year.
To clean fretboards, I use an old cotton bathing towel, cut into 5 inch squares and wipe between each frets. It removes most of the dirt without damaging anything.
For further cleaning, while oiling the board, I run down between each frets with the same 5 inch cotton towel to get excess oil out and to remove any gunk that manages to stay there.
The oil also helps keep frets from tarnishing. As with everything, find a method that works best for you. Have fun!