To open in the same window, you'd need to modify the /modules/mod_sclogin/helper.php file in the following line:
$html = '<a target="_BLANK" href="' . $profileURL . '">' . $html . '</a>';
Just remove the target="_BLANK" portion. I added an issue to make only external URLs (to Facebook, Google, Twitter, LinkedIn) open in new windows. On-site links should stay on-page.
For the avatar image, it looks like we don't have a good style around that image. However, the following in your CSS should do the trick:
.sclogin img {float:right}
Try that and let me know how it goes, or if you run into anything else.
Thanks,
Alex