9 speed cassette and 7 speed shifter
Option #4 won't work - the spacing between the cogs is different for 7,8,9 speed (4.34 mm for a Shimano compatible cassette for 9 speed, 5 mm for 7 speed Shimano). An index shifter is designed to move the chain over by a certain amount based on this spacing, which is why you have to typically match the number of speeds on the shifter with the cassette. Note that a 7 speed, 8 speed, 9 speed chain can usually be used on any of the 7/8/9 speed setups though (not necessarily optimally, but it usually works out with a bit of adjustment). 

The cheapest solution is to just buy a 7 speed cassette and put it on, and is likely the best (#2). 7 speeds chosen appropriately is more than enough for almost everyone. 

Disassembling a cassette is a hassle and its likely you won't get the spacing correct anyway, so #3 is out. 

Option #1 is a decent option given 9 speed alivio or whatever shifters are not too expensive now, and you get a fresh set of cables if you buy a set of Alivio shifters. But, if you have brake-shift combined units, you may also have to buy new brake levers. Its also more work, but if your shifters aren't working too well, this is a good choice - there isn't much on the market for good quality 7 speed shifters anymore, so if you're going to be using the bike a lot, this may be a good choice (I just upgraded my 7 speed commuter to 8 speed because the shifters went). 

To conclude, option 2 (installing a 7 speed cassette) or option 1 (installing 9 speed shifters) are the only ones which would work. Option 2 is cheaper (and I'd likely go for this if the shifters are in good working condition), but option 1 is a decently reasonable one as well (and is recommended if it looks like the 7 speed shifters will go).