Hi
With regards to garlic bulbs/cloves, Home grown or non chilled bulbs is best stored somewhere where there is good ventilation to prevent mould, fairly dark as the light encourages the garlic to sprout and the air does not want to be moist as this adds to the chance of mould so not the fridge!!!! Garlic should keep for 6 to 12 months like this as long as it is not supermarket bought garlic as it has usually been chill stored which causes early sprouting when exposed to room temp.
Not sure about the science of storing in oil. I keep my garlic ginger paste for no more than 4 weeks in the fridge as I have noticed the taste change after this time.
Regards
Barry