This is an excellent question, but one that deserves a vague answer. Actually, there is enough to see to spend part of the morning into the afternoon for the best of both!

Sorry I am totally ignorent about this language

Many people use the efficient, inexpensive, and frequent high speed trains throughout China. It is true they sell out, but I have never had a problem getting a ticket the day of going anywhere. I think that would depend on when you are traveling. Do not go during a Chinese holiday, don't travel anywhere in China during a Chinese holiday. I am not sure about Xian to Luoyang, maybe Xian to Zhengzhou, but buying online is your best bet, do so in advance and make sure you are on time, or you won't get a refund. Our friends found that out the hard way. Longmen Grottoes are the most beautiful thing I've seen in China, and I've been to 15 Provinces and 5 SAE (including Hong Kong and Macau). Have fun!

Anywhere between 2 & 4 hours depending how quickly you like to walk. Lots of stairs/steps up involved, so you need to rest once in a while

Terracotta is well known, just print a picture and show then you going here.

I'd think either spring or autumn from a temperature perspective. It can get very crowded so you have to take that in to consideration as well.

I'd agree, 3 to 4 hours if you find these kind of sculptures of major interest. The minimum would be an hour and a half, that's if you take the bus to the entrance (Not the visitors centre) and don't spend much time at each grouping of statues or the East bank. But I'd personally recommend leaving at least 3 hours.

We went there via fast speed train from Xian.

I had no problem getting a taxi. Yes there were a dozen taxis waiting.