With respect to the small steel splinters . I have had success in using the doughnut shaped strong magnet out of an old microwave .. placed it in a well fitting plastic bag , cable tied off the opening and placed it close to the work piece . It pulls most of the millings onto it . Before I take the piece out the lathe I spray it with WD 40 to dilute any cutting oil and work the piece with a clean paint brush this takes most of the crud off .
Getting the crud off the plastic bag is simple undo the bag over a plastic dust pan and quickly shake the magnet out onto a padded bench away from the dust pan the filings fall off the bag into the dust pan as soon as the magnetic influence decays.
Then I take the piece out the lathe & then use an old microfibre dish cloth/duster etc to wipe the item dry .
Using thick oil does indeed allow the millings to form a scum of needles so to speak .. keeping the cutting oil as thin as possible helps reduce the build up .
Re the micro fibre cloths ..
We tend to replace all 12 in the home each month as we use them as bathroom face cloths , polishing dusters & kitchen dishcloths/ table top wipers We have a different colour for each task. When new they are really soft and pleasing to use , by the end of each month I'm usually looking forward to a new set of washed & dried used microfibre cloths for playing in the garage with .
Even the older used ones still have a cushioning effect and tend to mask any sharp millings or turnings.
However using the WD40 to flush & a paintbrush to loosen first , then another WD 40 flush is the way to go each & every time for all metals .