According to the rules of the Premier League (PDF), the two teams are tied if the positions are irrelevant (i.e. no relegation or cup qualification is affected):
C.7: Subject to Rule C.14, if any 2 or more Clubs have scored the same number of points, have the same goal difference and have scored the same number of goals in League Matches in that Season they shall be deemed to occupy the same position in the table.
In listed tables for this case, the tied teams are typically listed in alphabetical order.
If the positions are significant, the teams play off for the position:
C.14: If at the end of the Season either the League Champions or the Clubs to be relegated
or the question of qualification for other competitions cannot be determined
because 2 or more Clubs are equal on points, goal difference and goals scored, the
Clubs concerned shall play off one or more deciding League Matches on neutral
grounds, the format, timing and venue of which shall be determined by the Board
Another regularly used-system (e.g. in European cup competition) is results between the teams that are tied on points (as you mentioned in part 1). e.g. In a league with Team A, Team B Team C and Team D, if Teams A B and C have the same number of points, a hypothetical league involving only fixtures between the tied teams is used to establish the relative ranking of those teams.