Днем Хануки
Dnem Hanuki
Happy Hanukkah = Hau'oli Hanuka
You shouldn't. Hanukkah and Christmas are no necessarily on the same day, so it would be inappropriate to wish someone a happy Hanukkah on December 25, if that day isn't during of Hanukkah. A better idea would be to look on a calendar to see when Hanukkah falls, then then wish your Jewish friends a Happy Hanukkah on Hanukkah.
"Tanoshii Hanukkah." It's a literal phrase and is likely to be gramatically incorrect.
Hanukkah is a holiday. Not a language. During Hanukkah you would say "hello" the same as on a regular day. You can also greet people by saying "Happy Hanukkah".
There is no such thing as "Hebrew Orthodox". There is only Hebrew. In Hebrew, you usually do not mention the name of the holiday in greetings. So you would just say Happy Holiday, which is hahg same'akh (חג שמח) Some communities do say Hanukkah Same'akh (חנוּכה שמח) meaning "Happy Hanukkah".
You say "Happy Holidays". If the person is Christian you can say "Merry Christmas", if the person is Jewish you can say "Happy Hanukkah" or if the person is African you can say "Happy Kwanzaa"
Счастливый День матери is how you say happy mother's day in Russian.
Not very common thing to say, but it's "hyvää hanukkaa".
In Russian, you would say "счастливое аппетит" to mean "happy appetite".
Happy Hanukkah= Boldog Hanukka
In Russian you say "Vy delaete menya schastlivym".
Just like this: Happy Hanukkah