The Eshel Avraham from Buchach (OC 260:1) makes almost exactly the points that you do in Q1. Here are 2 separate quotes from him that although unrelated, both apply:
נראה שאין גם מצוה מן המובחר לרחוץ כל גופו בחמין דוקא כל שרוחץ כל גופו... ...בצונן, וכן הוא מנהג כל עם בני ישראל
...כי עיקר הרחיצה היא משום תענוג, ולא כשהוא ענוי לו...
Regarding Q2, see Shemiras Shabbos Kehilchasa (old version 14:1:fn3) where this is discussed regarding what is considered hot water for washing on Shabbos itself, but certainly applies to our case (See the Biur Halacha inside, he brings a proof from this case). As usual, it is a Machlokes Haposkim, but the תהל"ד (תהליה לדוד) סימן שכו makes the point that obviously it is not referring to Yad Soledes Bo water, and suggests that if you can feel any heat it would qualify, although others disagree. Shemiras Shabbos concludes that according to Acharonim it is less than Yad Soledes, but does not give a precise definition. (See also Aruch Hashulchan, Dirshu Mishna Berura, and Piskei Teshuvos to the beginning of OC 326, where they bring a large number of sources that discuss the definition of "hot water" for Rechitza Beshabbos.)
Note that in both cases, I cannot prove that the Biur Halacha agrees, but I can provide other approaches within the corpus of accepted Poskim.