that white crust looks very unsavoury, do the clean out and wipe with bleach, then clean out the system. Yours is open-vented so most likely has sludge in it, so a good clean will do nothing but good.
1) find your drain cocks
2) tie up the ball float in the feed and expansion tank, bale it out, sponge it clean, wipe all. internal surfaces with bleach, including the float and lid. wipe dry
3) drain off at least a couple of buckets from your drain cock, more if it is bringing dirt out.
4) add a litre of sentinel x400 sludge loosener and then untie the ball float so the chemical is drawn down with clean water as the system refills. run it as normal for 4 weeks before draining and rinsing.
You will know the X400 is working if you bleed the radiators and find the water has gone jet black, this is loosened sludge washing around in the water.
Buy a bottle of Fernox Universal Biocide, you may have to order it from a plumbers merchant. You will add this to the F&E only after you have finished all your cleaning, draining, filling and inhibiting work, as it is fairly expensive. It will prevent future fungus or bacteria in the F&E.
If you can afford £100 and do basic DIY plumbing, add a system filter such as
http://www.screwfix.com/p/sentinel-system-filter/53852?_requestid=1841567 or
http://www.screwfix.com/p/fernox-tf1-total-filter-22mm/84311?_requestid=1841817 or the new Spirovent to a return pipe near the boiler, it will capture residual particles and prevent them forming a new blockage.
As for the dripping ballcock, a new one is so cheap it will be easier to swap it. You can usually leave the old stem in the side of the tank and just change the working mechanism by undoing the big nut. You can bring the old one downstairs and use a repair kit, then put it in a plastic bag by the tank ready to swap again next time. Adjust it so the water depth in the F&E is only a couple of inches above the exit pipe.
A brass one should last longer. You need a "part 2" with the plastic bridge on top. Pegler is good.
http://www.screwfix.com/p/pegler-prestex-float-valve-part-2/27627