With regard to the grout, there is a "grout pen" thing you can get in most hardware/diy shops. It basically resembles a tip-ex pen. You write it along the grout you have and it'll whiten it up. I don't think I'd recommend it as a long term job though.
Sealing the bath, use a good quality silicone sealant (you won't pay any more than 5 or 6 euro for the best quality one in any shop)....make sure its one with a milddew/mould resistant additive in it. Be sure and get out all the old silicone first though. There are a couple of good products on the market for that.
Cleaning tiles, I'm not 100% but I think there is a product by HG or Hagassen for tiles. They have a wide range of cleaning products, all of which are generally very good. Like I say, I'm not certain they do a tile cleaner, but I think they do.
Hope that helps