"Trains are fast."
How do you get your IME to type 速い? Cycling through the options given for "hayai," it doesn't come up for me.
Not necessarily, actually. There is a discussion elsewhere, and it's not clear cut. However, as it happens, I would prefer the other kanji here, you're right.
It is, however adjectives are treated very similarly to verbs in Japanese, including being able to end a sentence.
How do you say "The train is early" in Japanese? Since 早い (early) and 速い (fast) are both spoken はやい, do you say it differently to make your point clear?
I looked this up. This seems to be spoken as きしゃ and means a steam driven train. I am not certain if this can be used as train in general, then you're right. If not, well, most frequently you will find electric trains, aka 電車, and maybe on some not electrified rails there might be a diesel train.
Over 50 years ago, we Japanese called 汽車 as a train in general.
Recently, I think that only pretty elder people call it so.