AirDroid is what I use to mirror notifications and SMS. I've tried numerous variants like MightyText, Pushbullet, etc but this is the only one that's completely free for SMS purposes. The pro version doesn't offer a whole lot more.
I just noticed that I had to use MMS and not SMS. I have Cricket so not sure if that's different for other carriers or not. I was able to send to my E-mail from my phone texts and images and from my E-mail client Thunderbird I was able to reply and send. Just create the contacts in your phone and E-mail client and you're all set. No BS program or App to use.