We've settled that "requested" is more formal than "ask", but we didn't address the fact that "She asked me to stay" explicitely means that she asked the person who is speaking, whereas "She requested that I stay" does not necessarily mean that. "She" could have let someone else know that she wants "me" to stay, or she could have been asking someone else for permission.
Honestly it depends on who "she" is. Like, if I heard out of context "she requested that I stay" I would assume it's...like my boss requesting that I work overtime. In no "personal" situation would I ever expect say two friends to say something like "yeah, my girlfriend requested that I stay over tonight." That sounds very awkward. If your question is literally "does it mean anything else" then the answer is no...it just sounds weird to natives if you use it like that.
"She asked me to stay" is correct. It is, however, common to use the subjunctive after verbs like request, demand, recommend, ask, insist and other similar verbs. AmE and BrE differ in this respect.
• The doctor suggested that Joe rest. (AmE)
• The doctor suggested that Joe should rest. (BrE)
AmE has maintained the classic subjunctive while BrE uses the modal verb "should" plus the base verb. According to British linguists Chalker and Weiner: "the mandative subjunctive has made a considerable comeback in British English in recent years, probably under American influence."
Correct me if I am wrong:
"pediu" -> "pretérito perfeito do indicativo" (ação aconteceu e terminou num instante)
"ficasse" -> "pretérito imperfeito do subjuntivo" (dúvida, incerteza... o tempo combina com o verbo principal "pedir")
Comparando com os outros tempos verbais:
"Ela pediu que eu ficasse" - pretérito perfeito do indicativo + pretérito imperfeito subjuntivo
"Ela pede que eu fique" - presente do indicativo + presente do subjuntivo
"Ele pedirá que eu fique" - futuro do indicativo + presente do subjuntivo
"requested" -> simple past (the action happened, and it is over)
"stay" -> simple present (the doubt is subtle, and I am not sure if it really exist here)
Comparing to the other tenses:
"She requested that I stay" - past + present
"She requests that I stay" - present + present
"She will request that I stay" - future + present
I come from Australia (that other place where we speak English even though we don't border the Atlantic ocean (weird I know). 'She asked that I stay' sounds perfectly fine to me but ''She asked that I would stay' not only sounds redundant (as you noted) but wrong to me. 'She asked if I could/would stay' is also correct but it doesn't seem to work with 'that' instead of 'if'. However, my PhD isn't in English literature so I would value your opinion on this point.
Right. I wouldn't use "would" after "asked that" either. I don't have a Phd in Eng Lit, just as BA, but I have had a lot of experience teaching ESL - at home and abroad.
That said, I have learned spent more time thinking about the more arcane points of English grammar since "relearning" Portuguese on Duolingo and also doing the "tree" for Portuguese speakers learning English. Brazilians come up with a lot of interesting questions that have made me think more about linguistics and the "why" behind English grammar and usage of tenses.
I agree that "She asked me to stay." is more "normal" for American English. What I don't understand is the use of "ficasse" in Portuguese instead of "fique". That seems to transliterate to "She asked that I stayed." in the past tense. I am still trying to fully understand subjunctive usage in Portuguese.
The subjunctive verb in subordinate clause in Portuguese is determined by the verb in the principal clause. "Pediu" requires the past subjunctive 'ficasse'.
The subjunctive mode in English works differently. The subjunctive verb does not reflect for tense. It's the base infinitive (infinitive without "to").
• Jim insists that I leave. (Jim insiste que eu saia.)
• Jim insisted that I leave. (Jim insistiu que eu saísse.)
Since most of us communicate in informal situations, the infinitive (indicative) form is used more; however, the subjunctive form certainly has its place in AmE. As OrchidBlack (above) posted "they are used pretty much interchangeably."
I use both forms but would use the subjunctive mode in writing and in formal settings.