Since Deut. 22:25-26 equates rape with murder, would a man be required to give up his life rather than commit rape?
THIS IS ALL PURELY THEORETICAL!!!!
In broad terms, that's precisely the Talmud's argument. Though I think you mean verse 26.
Talmud Pesachim, 25b:
דתניא ר' אומר (דברים כב) כי כאשר יקום איש על רעהו ורצחו נפש כן הדבר הזה וכי מה ענין רוצח אצל נערה המאורסה הרי זה בא ללמד ונמצא למד מקיש רוצח לנערה המאורסה מה נערה המאורסה ניתן להצילה בנפשו אף רוצח ניתן להצילו בנפשו ונערה המאורסה מרוצח מה רוצח יהרג ואל יעבור אף נערה המאורסה תהרג ואל תעבור
(Deut 22:26) "Just as if one man jumped up and murdered his fellow, this [rape] is the same way." Why is the verse comparing murder to the violation of a betrothed maiden ... deadly force can be used to stop either, and one is obligated to be killed rather than transgress either.
Note, however, that Deuteronomy there was speaking specifically about rape of a betrothed (i.e. married) woman. The concept of "be killed rather than transgress" is limited to certain categories of prohibited relations. In this case, it's actually the adultery that generates the "be killed rather than transgress" clause, rather than the assault.
It's less clear, then, whether assaulting a single woman would be in the same category. (Mind you, trying to assault anyone could very likely wind up with the death of the assailant, victim, or both, which would invoke the rule of "be killed rather than murder.")
In Yoreh Deah siman 157 the laws of which sins a person must give up his life so as not to transgress are discussed.
In siff one, in the last gloss of the Ramma he points out "and any issur of idolatry, or Giluy Arayos (sexual misconduct), and murder, even though the sin itself does not incur the death penalty but is only a 'lav', one must give up his life in order not to transgress it".
The Shach #10 brings a gemara from Sanhedrin 75a concerning a man who became obsessed with a certain woman, and the doctors said he would die unless he had relations with her. The Chachamim ruled he should die. How about for her to stand naked in front of him? Die. Talk to her from behind a fence? Die.
The Shach quotes the Ran and other sources who point out that these sins would not be real Giluy Arayos, but are included in לא תקרבו לגלות ערוה, which is lav of Giluy Arayos.
He goes on the use this gemara to rule with the Rambam who said kissing and hugging is included in the lav of lo sikrivu, and against the Ramban who said they are not.
The Beis Yosef on that siman brings the Rambam in chapter five of Yisodei Hatorah halacha nine, who rules with the opinion in that gemara that we are dealing with a single girl, not an eishes ish. The Beis Yosef gives a reason and a proof from the gemara to the Rambam's ruling.
So all in all, rape, even of an unmarried girl is a sin that one must give up his life so as not to commit.
If the victim is one of the real Arayos of the Torah, such as a nida, a married lady, a forbidden relative, a male, or animal, than this would be even more universally accepted, but again, we don't rule with the opinions who made that distinction anyway.