I will be on a business trip to Romania but I have plans to have a vacation in Paris and Italy. Can I apply for a Schengen visa while I'm in Romania? I'm from The Philippines.
No, you cannot make a Schengen visa application in Romania without being a resident. Relevant quote from the Czech consulate in Romania:
The applicants file their application in the state of which they are either citizens or holders of a long-term residence permit. In case there is no diplomatic mission of the Czech Republic in this state, the application is to be filed at the diplomatic mission which is accredited for that state (usually located in a neighboring state) or at the diplomatic mission of another Schengen Area State if the Czech Republic concluded a representation agreement with this state.
Generally speaking, no. As Tor-Einar pointed out, you have to apply at the consulate for your place of residence. And since Romania is an EU member, the French embassy in Romania won't have the staff to process your application.