Your situation involves personal contact with an object in air (gloves, hands etc. which is not a risk for hiv. No worries, because you can't get hiv from personal contact except unprotected penetrating vaginal or anal, neither of which you did and you didn't share hollow needles to inject with which is the only other way to acquire hiv. Analysis of large numbers of infected people over the 40 years of hiv history has proven that people don't get hiv in the way you are worried is a risk.
HIV is a fragile virus in air or saliva and is effectively instantly dead in either air or saliva so the worst that could happen is dead virus rubbed you, and obviously anything which is dead cannot live again so you are good. Blood and cuts would not be relevant in your situation since the hiv has become effectively dead, so you don't have to worry about them to be sure that you are safe.
There is no need to sanitize gloves that touch hiv.
I have to fully agree. This is not a risk for HIV. I think if you are in the day to day job of caring for people though, it would be important to learn proper usage or protective wear, proper steps to take if you have an issue and want to inquire about it with your manager, and management of your own anxiety. If you tend to worry a lot, this may not be the best job for you. Treating anxiety is important. The only ways adults get HIV is from unprotected vaginal or anal sex or sharing of IV drug needles. Needle sticks do not cause transmission. That is different than injectable needles.