(1) If there is no option left to maintain good relation between spouses, then it is lawful to divorce in compulsion. The preferable method of divorce is that first you should divorce only once, then wait for one month. If you feel that separation is inappropriate you can take her back in iddah period; otherwise leave her she will automatically turn baina (separated by talaq) once her iddah period is over.
(2) Yes, it is necessary for woman to pass iddah after talaq. If she does not observe iddah then too the iddah shall be fulfilled and talaq shall be counted valid. A woman who does not pass her iddah period shall be sinner. It is necessary to do taubah and istighfar from it.
Allah (Subhana Wa Ta’ala) knows Best
Darul Uloom Deoband