How long does beer keep if fermented and kept in a bucket?
My OH keeps his beer in a bucket for 10 days then bottles or barrels it. Any longer than that and you risk the dead yeast cells causing autolysis ('off' flavours from the breakdown of dying yeast).
If taken out of the bucket to store, which is better bottling or bag in a box?
Again OH kegs his, you can buy fairly cheap plastic barrels for around £20 and if you are storing for any length of time glass bottles is advisable over plastic as the plastic leaches out Co2 over time. Your glass bottles should be either dark themselves or keep them in a dark cupboard to avoud 'light strike' which is when the UV starts breaking down the flavours in the beer.
If bottling what are the pros and cons of grolsch bottles vs bottles with crown caps?
Re-usability is the key, the Grolsch bottles are great for re-using but require a bit more work to sterilise whereas crown caps are easy to sterilise but need to be bought every time.
Beer in a bottle or keg can easily be kept from anything between 6 months to a year and the longer you leave it the better it will taste.
Easiest thing for you to do would be to get a starter kit and depending on wheather if comes with a barrel or a bucket buy the item that is mising independently.
If you were thinking about all-grain brewing though I would advise you get some kit brews under your belt first then reseach all-grain brewing as a) it's quite complicated and b) it's very expensive to start out.
Give us a shout if you need any further advice my OH is a walking brewing handbook
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