You should generally get your foot drop treated by a neurologist (a specialist in brain and nerve disorders).

Whether a foot drop can be treated or not depends on its cause. A treatable cause of foot drop is curable, but if the root cause is not treatable, then foot drop is not curable. So sometimes, the foot drop can persist permanently.

Yes, physiotherapy can help in the treatment of foot drop as:
It strengthens the muscles of your leg
It provides an excellent range of motion in your knee and ankle
It helps in the improvement of associated gait problems
It supports movement at the heels due to stretching

Yes, surgery for foot drop varies as per the cause:
Nerve surgery for newly developed foot drop
Surgery promoting fusion of foot or ankle bones or an operation involving the transfer of a working tendon and attached muscle to various parts of the foot in old cases

Some home remedies helping in the management of foot drop include:
Using a brace (fitting ankle and foot)
Splint supporting your foot by fitting it in your shoe
Regular exercise and stretching that helps strengthen leg muscles
Some basic rules to be observed to prevent fall, such as proper lighting of the house, moving away obstacles from the floor that increase the risk of fall, etc.