Minimum 03 days you need,
One for Nathula and Baba Harbhajan Singh Mandir, War Memorial, Tsogmo Lake
Second day for Namchi and Hanuman Tonk/Mandir
Mall road and local sight seeing
Sikkim is itself best. but for traveling, gangtok to North sikkim or silery gaon to silk route. also ravangla namchi and a lot more beautiful hamlets to explore.
Same has happened with me earlier they used to approve it in couple of days however this time they are taking more time, I guess it'll be done in a week or so.
Both stations are there in the rishikesh itself ... book tickets to any of them and you may reach rishikesh ... you may get a local auto for 50-100 bucks to your hotel anywhere in rishikesh after that ... RKSH is near TRIVENI GHAT ... YNRK is a little afar !
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