Fixing a sink with multiple problems
First thing is close the saddle valves you mention in the other room. The cold on IS on right? Maybe someone shut if off because it was leaking? Open the faucets to relieve pressure and leave them partially open to facilitate unscrewing the shafts once the handles are removed.

The handles should pull straight up once the screws are removed. If they do not come off relatively easily try prying them up with a big screwdriver and some kind of fulcrum. If they still don't budge they are likely very corroded onto the knurled shaft. In that case try soaking them in CLR, a readily available cleaning solution for removing calcium, lime and rust. It's a mild acid solution. If they still don't come off you may have to replace the whole assembly. I just went thru this with my old shower faucets. I couldn't even get them off with a gear puller and ended up destroying the handles just to get the assembly out of the wall. In your case I actually would recommend replacing the assembly anyway as they're relatively cheap and the newer cartridge style faucets work much better and require less maintenance. But if you want to persist....

Once you get the handles off remove the bonnet (big nut and you circled) and you should be able to unscrew the knurled shaft and washers in the direction that you would tun on the faucet. You should also find either graphite 'packing" or maybe some other kind of replaceable rubber seal in the bonnet. The graphite packing string can be found at a good hardware store, (i.e. NOT a big box retailer) and if it's a real good hardware store there will even be an old guy there who can tell you how to repack the faucet. There should be instructions on the package too. Also replace the rubber washers to seal in the off position. 

Looks like some pretty hard water from the looks of that cold supply. First thing I'd do is disassemble the cold faucet and slowly and carefully crack the shutoff open on the cold supply and see if you're getting water thru that plastic supply. It will obviously come out thru the whole left when unscrewing the shaft so go easy. If no water then you have a blockage somewhere further back and you'll have to go backwards thru connections until you find a blockage. If you do get water there and still none once the faucet is reassembled then there's a blockage somewhere within the faucet and again just replacing the whole assembly would be the easiest solution.