Simple animations in Ionic Framework

Yes you can, have a look on this.

A powerful javascript animation engine for web and hybrid mobile apps, inspired by Facebook Pop, built by the Ionic team.

https://github.com/driftyco/collide

var animation = collide.animation({
  // 'linear|ease|ease-in|ease-out|ease-in-out|cubic-bezer(x1,y1,x2,y2)',
  // or function(t, duration),
  // or a dynamics configuration (see below)
  easing: 'ease-in-out', 
  duration: 1000,
  percent: 0,
  reverse: false
});

// Actions, all of these return `this` and are chainable
// .on('step' callback is given a 'percent', 0-1, as argument (for springs it could be outside 0-1 range)
// .on('stop' callback is given a boolean, wasCompleted
animation.on(/step|destroy|start|stop|complete/, function() {})
animation.once(...) //same event types
animation.off(...) //works like jquery.off
animation.stop(); //stop/pause at current position
animation.start(shouldSetImmediately); //start from current position
animation.restart();
animation.velocity(n) //starts the animation going at the given velocity ,relative to the distance, decaying
animation.distance(n); //distance for the velocity to be relative to
animation.destroy(); //unbind all events & deallocate

animation.isRunning(); //boolean getter

//These are getters and setters.
//No arguments is a getter, argument is a chainable setter.
animation.percent(newPercent, shouldSetImmediately); //0-1
animation.duration(duration); //milliseconds
animation.reverse(isReverse);

animation.easing(easing); //setter, string|function(t,duration)|dynamicsConfiguration.
// Dynamics configuration looks like this one of these:
// animation.easing({
//   type: 'spring',
//   frequency: 15,
//   friction: 200,
//   initialForce: false
// });
// animation.easing({
//   type: 'gravity',
//   bounce: 40,
//   gravity: 1000,
// });

Animations in ionic are just the same as if you were working on a standard html page, the only difference is how to set it up in angular.

It is within the controller that is assigned to a view that you can add your code.

.controller('homeCtrl', ['$scope', function($scope) {
    // Jquery animation
    $(selector).animate({params},speed,callback);

    // Greensock animation
    var photo = document.getElementById("photo"); //or use jQuery's $("#photo")
    TweenLite.to(photo, 1.5, {width:100});

    // or define a function and call it when you want an animation to run
    $scope.animate = function(element) {
        $(element).animate({params},speed,callback);
    }
}])

Thats the gist of it, you can look into more detailed stuff on YearOfMoo

Yes. Nothing in Ionic would prevent this. Did you actually try it? Or are you asking if Ionic ships with anything like that? As far as I know the answer would be no.