1) According to the Premier League Handbook, 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.17, if any two 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 two 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.
2) 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.