7-speed shifters won't work with 8, 9 or 10-speed cassette.
Also: 8, 9 and 10-speed cassettes are all the same width, while 7-speed cassettes are a bit narrower and fit on a narrower freehub body. You can use a spacer to take up the extra width when mounting a 7-speed cassette on an 8/9/10-compatible freehub body. I'd recommend eventually upgrading your cassette and shifters (rear der will work fine, no meaningful difference between shimano rear deraillers "designed for 7-speed" and those designed for 8/9/10 speeds).
One other thing: older 7-speed freehub rear hubs had 126mm spacing (distance between the locknuts) whereas 8/9/10-speed freehubs are 130mm spacing (135mm on mountain bikes, which was true for 7-speed mountain bike freehub rear hubs as well; but yours is a road bike so this doesn't apply). Your frame may have the rear dropouts spaced at 126mm, and you'd need to flex the frame a little to fit a 130mm-spaced rear hub in.
I'd recommend that you figure out the rear spacing of your bike's frame and freehub.
If your rear wheel is indeed a 7-speed freehub, and your shifters and cassette are in good shape and don't need replacing anyway, and you don't feel the urgent need for more gearing options, I'd recommend that you buy a rear wheel on eBay with a 7-speed freehub body.