Whilst there are big difficulties in removing scratches - also as Max says, glass that has become etched is often beyond the pale, to quote neil........'so I think a tide line to a certain extent'
. Generally, tide marks/lines can often be removed (sometimes as much as 95%) - when using chemicals, power sources and much patience. It does depend of course for how long the staining/etching has be working on the glass. I think you may be resigned to having to live with some of the problem, unless you do go professional, but it really is worth the effort to try and see how you get on, please do keep us informed. Often the worst culprits are the almost unseen deep scratches. Somewhere on the board (using the 'search') there should be notes to help you regarding removal of staining/scratches etc.
thanks Anne - I see neil is posting from south Hampshire
but maybe his glass spent most of its life up north.