He's my boyfriend (he's mine) so don't flirt with him, don't show off, and don't touch him.
「彼は私のだから触れないでいちゃつかないでアピールしないで。」は英語で「He's my boyfriend (he's mine) so don't flirt with him, don't show off, and don't touch him.」という表現が良いと思います。
触れないでーDon't touch
いちゃつかないでーDon't flirt
アピールしないでーDon't show off