The issue with text coming two lines at a time and out of order are issues I've had when texting from my phone with users of certain Android phones. The issue relates to the fact that SMS texts are technically limited to 160 characters, and anything longer than that gets broken off into a second message of 160 characters. iPhones and certain Android phones are smart about piecing the message together so that it just looks like one block of text, but others don't handle the process quite as well.
That doesn't necessarily mean that there's no issue for you with texting from your computer. It's just a possible explanation for what's going on, made more plausible by the fact that it's a lot easier to type long messages on your computer keyboard than on your phone 🙂