Is one allowed to request a refuah sheleimah (a speedy recovery) for oneself, while saying Shemona Esreh? If so, does one need to say one’s Hebrew name, or just say אותי ("me")?
Well this is exactly the content of the Refaeinu blessing
וְהַעֲלֵה רְפוּאָה שְׁלֵמָה לְכָל מַכּותֵינוּ
Bring complete healing to all our wounds
So you seem to be covered for yourself. If you wish to add the standard yehi ratzon addition, you can do as well: OU here says
If one wishes to ask special consideration from God for oneself or another who is ill, he may insert a personal request here. (Many siddurim offer a suggested text for such insertions.)
The Mishna Brura (SA OC 116:3) writes not to say the name of the sick person if one is in front of him or her. I presume this would apply to oneself.
As always, CYLOR before applying anything you read here.