The machines like the HQ Sweet Sixteen and the Tin Lizzie do not have feed dogs. So far as I know, none of the mid arm machines do. Therefore, you are limited in what you can do as they are designed specifically for Free motion quilting.
Theoretically, you could do some straight stitch sewing on them, but you have to move the fabric yourself, so you need to be really good at managing your stitch length since you would be FMQ all seams and the pieces need to be large enough for you to be able to get a grip on them while they're under the needle since there are no feed dogs to move the fabric. I've seen a couple of tutorials where people are using them in very limited ways for some piecing but you really have to be very good at Fmq to get good seams with consistent stitch lengths if you want to use them to stitch a seam.
Since you want a machine that you can Use for regular sewing and has a larger harp for FMQ, have you thought about the high end domestics like:
the Bernina 7 or 8 series, the Viking Sapphire series, The Brother Dreamweaver and the Babylock equivalent, or the Janome 7700, 8200, 8900, or the Elna equivalents. These machines all have 10" and 12" throats, regular sewing stitches as well as lots of decorative stitches and all do a good job at FMQ according to reviews I've seen. Many of them can be purchased new for the same amount of money, or less than the HQ Sweet Sixteen.
A lot of us here on the board have larger straight stitch only machines like the Brother PQ1500s, the Juki 2010' etc. these machines have feed dogs and are straight stitch only machines. They also do a beautiful job of FMQ and have a harp that is 9" to the right of the needle and 7" high which gives plenty of room for quilting most quilts once you learn how to manage the bulk.
All that said, if it were in my budget and if I had space for another machine, I'd love to have one of the sit down (move the fabric) quilters like the HQ Sweet Sixteen or the new one that Juki makes with an 18" throat, or the APQS George. I've played with all of those and because of the much higher space in the throat, they have much better visibility around the needle than do any of the large domestics, which helps cut down on neck tension if you're doing things like ruler work where you have to see behind the needle-- with those machines you don't have to crane your neck around the machine to see behind the needle. To me that would be the main consideration for buying a mid arm, which to my mind would eliminate the Tin Lizzie--- as I recall it doesn't have the vertical space that the others have.
Rob