You can definitely do all of that with Construct 2.
Parts 1 & 2 are the most simple, as all you have to do is create some local variables for each statistic or value that you want to track (I typically do this by creating an object in the layout view and then adding one from the list on the left), and then after each fight, you add the experience that they got, see if they leveled up, and if they did, then give them X equipment.
You're going to need to use something like AJAX to connect to your server though and store the data (here is a tutorial on that scirra.com/tutorials/61/ajax-example-with-construct-2). Once you retrieve the data though, you just have an automated system run through the fight (typically a loop until one player is dead, with whichever attack patterns and calculations you want in between).
Hopefully that helps!