Vocabulary in The Love Song of J. Alfred Prufrock
Death personified. A regular footman is a servant who generally admits and waits on guests--the suggestion being that if Death is holding your coat, then you've entered a place where either you're not likely to come out or are on Death's doorstep.
These are establishments that have thrown sawdust over the floor to soak up spilled beverages and liquids. They're inherently seedy or low-class restaurants and are associated with "cheap hotels" and their sundry clientele. This walk indicates that the poem's speaker is familiar with the red-light district, either because he lives nearby or because he frequents it for other reasons.