What is the ruling on a woman using a public platform like "ask" to answer purposeful questions and using emoticons, while fearing falling into errors, knowing that she refuses to answer questions from men?

The default ruling concerning things is permissibility and lawfulness until evidence of prohibition arises. Using the "ask" website is permissible for men and women as long as there is no Shar'i (religiously lawful) impediment in it, such as forbidden communication or the spreading of forbidden speech. Emoticons are not forbidden in their origin, but a woman should not use them in conversations with men due to what they may lead to of temptation (fitnah).