I will be in nanjing in November and like to visit luoyang for a day.

1. Yes, the time is too tight. It takes around 4 hours by high speed train from Nanjing to Luoyang first.
2. It depends on you. The city center is also good.
3. To Longmen grottes:You can take bus V9 to it with around 50 minutes and CNY1.
To the temple: You can take bus d3/3 to East Luoyang Station, and transfer to bus No.801 to white horse station. It takes around 1.5 hours with CNY5.

hi im from Luoyang, I think your schedule is little intense, because from Nanjing to Luoyang is around 4-5 hours by bullet train. you may live in Luoyang for one night, you can go to Luolong dist. by taxi( 10 mintues, 10rmb around) to find hotel, where is clean and new. longmen grotto is far away from white horse temple, and luoyang is pretty hot now, you can go to white horse temple first and then go to longmen. longmen grotto is large if you plan to walk, it may takes you half day, you can also travel longmen at night, there is light show at night,
Transportation is cheap and convenient in Luoyang, you can travel by bus or green taxi,
Hope this answer is helpful, have fun!