Sorry to her this. Advice for what it's worth.
1. Nausea: Take Zofran ahead of the infusion or ask for IV Zofran. Keep Zofran tablets with you 24/7. The moment you feel nausea, pop a tab under your tongue and allow it to dissolve. The nausea, in my experience, stops and resolves within about 5 minutes.
2. Doxorubicin (Adriamycin) is a rather nasty drug and is well known as a cause of redness/skin rash. Lacking constitutional symptoms of allergic reaction (sneezing, congestion, watery swollen eyes etc.) it is 'probably' the anticipated side effect of the drug.
In any case, do let doctor know so that they are aware of it. If it becomes problematic, your dose can be adjusted.