2.
to be precise,
一月は最初の月(です)。
ichigatsu wa saisho no tsuki (desu).
mutuki(睦月) is an age-old word means Januray in Japanese.
off course "ichigatsu wa mutsuki" gets across enough as Japanese.
3.
calb0y is right.
in addition, a way of saying friendly.
一口、ちょうだい。
hitokuchi choudai(or, cho-dai).
4.
I'm not intimate as for English, but if you express the sentence like "the victory was got by myself" in above phrase "the victory was single handed", I translates freely that
(わたしは)勝利を、この手でつかんだ。
(watashi wa) shouri wo kono te de tsukanda.
nominative is often omitted in Japanese.
or, if the sentence like "the victory was held single handed",
(わたしは)勝利を、一手に握った。
(watashi wa) shouri wo itte ni nigitta.
and to Mirage, translating "The prince was one step ahead of the princess." to Japanese literally,
王子は、王女より一足、先にいた。
ouji wa oujo yori hitoashi saki ni ita.