"Làm ơn nói tiếng anh" or "Xin hãy nói tiếng anh" is the right way :D "Họ mừng cho cô ấy" is how we say it in Vietnam, formally. That sentence above ("Họ hạnh phúc cho cô ấy") is very unnatural to hear. I've never heard anyone say it. AND even "Họ mừng cho cô ấy" is rare, I often think about it as the way a hypocrite congratulates someone, since it's too formal and doesn't sound "happy for her". Some same-same variations are "Họ chúc mừng cô ấy" or "Họ thấy mừng cho cô ấy" (this is the way I would say). The most natural way in my opinion is "Họ thấy mừng cho cổ" ("cổ" = "cô ấy" in Southern Vietnam), as I live in the southern part of Vietnam.
Yeah, "Họ chúc mừng cô ấy" is "They congratulate her", and it's not really the same as "They're happy for her". That's why I said it's only a same-same variation. "Họ thấy mừng cho cô ấy" or "Họ mừng cho cô ấy", on the other hand, means literally "They feel happy for her", which is in my opinion close enough to "They are happy for her".