When doing this course on PC, I have a table where I have the hiragana and katakana saved to "copy & paste" them - which somehow feels like cheating. But alas, my keyboard has no Japanese symbols. But for my mobile phone I can/could download and install keyboard layouts for Japanese amongst others. Maybe, if you're using a mobile phone, you could look if something is available. (at least for my Windows Phone it is offered) But I am not sure of how to type Kanji "directly", too.
On Windows 10 I set up japanese keyboard by going to "Settings" -> "Time and Language" -> "Language" and added japanese as a language with "Microsoft-IME" keyboard.
With this you can just type japanese syllables using the latin letters on my keyboard. You can switch between Hiragana and Katakana with predefined shortcuts and the Kanji symbols are just offered when typing Hrkt (Hiragana/Katakana).
For Android (phone, tablet and such) users, there are a couple of ways to use and install a hiragana keyboard;
One way is to have gboard updated to the latest software: go in to keyboard settings then find the language input, to go in to the settings you can find the gear icon on top where the next word bar is at then click on the google icon, it'll open up keyboard settings then I am sure you can figure it out from there.
The second option consists of going to google play and searching for the hiragana keyboard.
It'll automatically suggest fitting/common kanji. For example, it would suggest 日本 from にほん. More obscure kanji are a bit of a pain in the ass, and since romanizations are bullshat at best with duolingo it can be hard to get how you input hira/katakana to actually be what it should be to get the suggestion.
Just a quick explanation on hiragana, katakana, and kanji: Hiragana and katakana are both phonetic(sound) alphabets. Hiragana is used for Japanese words, while katakana is used for foreign words (ex. アメリカ). Kanji are characters that each represent an idea and have multiple pronunciations (but you'll learn those later).
Traditional Japanese names are generally written in kanji; especially family names or in a formal context. You may see given names written in hiragana though, because it is nearly impossible to guess how the kanji of some names are pronounced. (People often use kanji for the meanings and add entirely new readings to them for names). So hiragana may be used instead in order to clarify pronunciation. It is also not uncommon to see them in katakana, especially with kids/teens as it is seen as 'cuter' and similar to writing in all caps in english.
Foreign names are generally written in katakana as that is the script used for loan words. Some foreigners living in Japan may adopt Ateji kanji though, Kanji chosen by them that hold the same pronunciations as their name. This is far less common but it is similar to when an asian person adopts a western name when living abroad.
in fact: the Japanese language consists of Kanji as well as of Hiragana and Katakana. Both, Hiragana and Katakana, are "syllable alphabets" out of which Katakana is used for words which are not originated from within Japan (like words borrowed from English or people's names for non-Japanese [people]). Theoretically, it would be possible to "substitute" the Kanji with Hiragana. But "in the real world" you wouldn't get too far, with only the knowledge of Hiragana and no Kanji. I tried to get my hands on Japanese with some small introduction units at a local "Japanese culture center". Hiragana and Katakana were ok to learn... but I couldn't get the Kanji into my head, to be honest.
Within Japan, you will often see the mixture of all three, even in one sentence: Kanji, Hiragana and Katakana. Sometimes there is a way to write the syllables of Hiragana/Katakana with the letters we use. This is called "Rōmaji". But I rather saw this in learning materials to help understanding the names and pronounciation of the syllables. And sometimes, when an uncommon Kanji is used, a translation in Hiragana stands next to it (sometimes directly above) so that the reader might be able to get the meaning.
But that's all I know about it for now. I hope that helps, Hafsa.E - I only "leveled" the Japenese language to level 6. I don't know if there are Kanji later on in the tree. (It might be possible, but since it is still in "beta"-version the tree might be still incomplete.)
It would be very strange for Japan to call themselves by a foreign word rather than their own native name...
You would mainly only really use ジャパン if you were talking to someone who didn't speak/understand Japanese well (like a native speaker trying to explain where they are from to a foreigner) or in a stylistic choice that may be more accessible to foreigners as well (in names of companies)