I feel there is a mutual and symbiotic relationship between the long and the short game. Practicing short game more has really helped me swing all the other clubs with better tempo and rhythm, bunker play I feel is especially helpful in this regard for some reason.
However, if you're a high handicapper, it would behoove you to practice more on your long game mechanics and technique in my opinion. Learning to use the correct mechanics and establishing proper contact goes a long way in lowering your handicap.
It's different for every player and what level he/she is at.
Agree 100% about option 2. The progress I've seen (from 110s to one round in the 70s in about 2.5 years) has been based on a lesson every 2 or 3 months (I've probably had about 13 lessons). The progress is generally as such:
1- something to work on. Talk heaps in the lesson about many ways of doing it (for me it has been mainly centered hip turn on the back swing, then about how to coordinate the hip turn etc).
2- practice, play, implement etc. Progress tends to start strong then level off (unsurprising)
3- when I no longer understand why the bad shots are happening go back to the lesson. This hasn't involved repeating more than 25% of the previous lesson, i.e. each lesson is at least 75% 'new' stuff.
This progress wouldn't work for weekly lessons. Wouldn't have enough time to make proper changes, would probably detract from the time I have to devote to playing, and I think would end up in the coach over-complicating things due to the activity bias of being involved in a lesson. Ie I'm here, better give me something to work on, even if I didn't need anything new.