I am assuming by your post that the lease is signed but the tenant does not have keys yet, keep it that way. Yes you have a signed lease but that lease also says that move in funds are paid but they are not so they are in violation. There was also clear fraud on the application front which is another strike to back out. Keep showing and ignore her threats and cease communication. People threaten to sue all the time but unless you hear it from a lawyer it is all smoke and do you really think they would win even if they did. Keep moving forward and ignore it.
That said there were mistakes made that it seems you are learning from. Move in funds need to be made in certified funds through a certified or cashiers check, without this there is no move in or keys. I have also accepted a wire from a tenant who did not have a local bank where they could get a certified check. I also have them send me a copy of their drivers license after the background check so I can verify their info before the signing. All said though you dodged a bullet and overall got a far cheaper lesson in screening and move in funds than others have.
I will disagree with the last part of your post though on hiring a property manager which no doubt you are going to get suggestions anyway you need one. Nobody is perfect starting out but really this isn't that hard especially if it is local and you don't have many properties. Keep your head up and keep moving forward and you will do great. While they work out great for some people nobody cares about your money like you do so don't let one setback scare you off.