"It's five o'clock."
Nope. Since Japanese doesn't use spaces, kanji is used to arbitrate meaning and provide punctuation (if you will). Things such as numbers, time, established Japanese surnames, etc should always be in kanji.
日本人 actually write numbers as numerals like the rest of us. They think it's time consuming and silly to write numbers in kanji when you can just write the numerals and everyone understands them - lesson I learned when I had just moved to Tokyo and thought I was clever writing my phone number out for a friend in kanji. If you write your numbers in kanji all the time be prepared for people to laugh at you.
What are these new that we are learning called? Kanji? Romaji/romanji ? Or something else?
Kanji, they are chinese characters and are also used is japanese. Romanji is just writing out how japanese sounds with english letters. (Ex: "go" is romanji for 5). It's just used for learning.
In every day speech, would the 時 be dropped out or is it always kept?