Do the arbiters have the obligation to write the moves when players are not doing it?

In general, "No". In rapid (30+0 counts as rapid in the FIDE Laws) there is no obligation for anybody, neither players nor arbiters, to write the moves. The only instance in which the arbiters would write the moves for a player would be in case of -

1. Disability. For some reason the player cannot write the moves down.
2. In case of a standard time control with multiple controls where there is no increment in the control and it is not the last control and the player has less than 5 minutes left. The only place these ever happen is elite events.

If players are not writing their moves can they claim a draw using the threefold repetition rule

There needs to either be a record of the game or the arbiter needs to be watching and paying attention. So, if the opponent is still recording because they have more than 5 minutes left then their scoresheet can be used to check the moves. If moves are being recorded electronically (live boards) then in some cases these can be used. Finally if the arbiter is watching and counting then they can verify the claim.

If the arbiter notices that a position has been repeated three times should they intervene if the players do not claim a draw?

No. It is expressly forbidden. This would be interpreted as helping one of the players. There is an additional rule which says that if the arbiter sees that there have 5 repetitions then the arbiter MUST intervene and declare the game drawn.

Do the arbiters have the obligation to write the moves when players are not doing it?

In general, "No". In rapid (30+0 counts as rapid in the FIDE Laws) there is no obligation for anybody, neither players nor arbiters, to write the moves. 

The only instance in which the arbiter or another person other than the player would write the moves for a player would be in case of -

1. Disability. For some reason the player cannot write the moves down.
2. In case of a standard time control with multiple controls where there is no increment in the control and it is not the last control and the player has less than 5 minutes left. The only place these ever happen is elite events.

If players are not writing their moves can they claim a draw using the threefold repetition rule

There needs to either be a record of the game or the arbiter needs to be watching and paying attention. So, if the opponent is still recording because they have more than 5 minutes left then their scoresheet can be used to check the moves. If moves are being recorded electronically (live boards) then in some cases these can be used. Finally if the arbiter is watching and counting then they can verify the claim.

If the arbiter notices that a position has been repeated three times should they intervene if the players do not claim a draw?

No. It is expressly forbidden. This would be interpreted as helping one of the players. There is an additional rule which says that if the arbiter sees that there have 5 repetitions then the arbiter MUST intervene and declare the game drawn.