there's no such ac (in pak) (only a few daikin variants available in some regions have eer in +- 4.3 range, otherwise all brands are in 3.1-3.6 range including daikin Mitsubishi Panasonic gree tcl midea samsung sharp toshiba haier etc)
EER has nothing to do with energy-saving % stickers
generally, no
LG, TCL etc offer amp lock mode, which is available in both remote and app
there's no such app
in universal, only IR-based apps are available which offer basic temperature and fan controls (if your phone has an IR blaster)
all brands recommend their own app, which offers better control and functionality (depending upon the brand, some apps are average in terms of UI and functionality, some are better)