I recently had a rectal exam, where the doctor put his finger in my back passage, and might have felt my prostate. I was not aroused, but when I later checked my pants, there was madhiy. I naturally produce madhiy randomly, and I do not know if this was because of the exam or from before. Do i need to do fard ghsul? (1) Doing fard ghusl is very difficult for me as I have OCD and spend very long flossing, and brushing, gargling and washing my nose, as I try to make sure the water reaches all parts of my mouth, and am not sure if my nose is clear of mucus, and my nose is often blocked. Is it okay to follow the shafi ruling which does not require washing the mouth and nose as fard of ghusl, because i am mentally ill. (2)
Ghusl is Fard when there is ejaculation with arousal. So in your case Fard Ghusl is not necessary. If there is Madhi discharge then the ruling is to wash it from the body and clothes is necessary and Wudhu will be required.
The Fard of Ghusl is washing and rinsing the whole mouth once and washing the nose once - as you normally do for Wudhu, there is no need to exaggerate in the washing of the mouth (i.e gargling) and nose but it is Sunnah to do so according to the Mu'tamad (relied upon opinion).
وفرض الغسل عندخروج مني منفصل عن مقره بشهوة أي لذة ولم يذكرالدفق ليشمل من المرأة ولأنه ليس بشرط عندهما خلافاللثاني
(الدرالمختارمع ردالمحتار,كتاب الطهارة, مطلب في سنن الغسل,ج.1ص.156)
وفرض الغسل الخ غسل كل فمه ويكفي الشرب عبالان المج ليس بشرط في الأصح وانفه حتى ماتحت الدرن