Is it legal to build an app that is built on top of another company's software?

You would have to have a freedom-to-operate study - also known as an infringement study - performed in order to determine if the IOSAndroid app is patented. If it is, then any use of that app, whether you are enhancing it or otherwise using it, would constitute infringement. If there is a patent, you may also want to have a validity study performed with respect to the patent in order to determine if that patent is valid - if it is, it is likely that you would infringe the patent, if the patent is not valid or you feel that it is not valid, you may not be infringing the patent. Consult a patent professional to have these studies performed.

Possibly. The "very popular social media iOS/Android app" itself will be protected under copyright and, potentially, patent(s). Depending on how your software will be interfacing, through the third-party API or otherwise, with the app may raise Stored Communications Act, Computer Fraud and Abuse Act, and privacy-related issues. If a problem does develop, the fact that you were using the third-party API will not likely prevent the app owner's claims from being at least asserted against you. A good answer to your questions will be very fact-dependent and it is thus very advisable to consult with local counsel in private before substantial development.