Effects of internet connection speeds on server queries

No. If the server is on the same machine as your phpMyAdmin install your remote machines bandwidth will not impact the performance of your queries.

As a side note, phpMyAdmin is the devil. There are lots of security vulnerabilities in it and it's rarely secured properly. You might want to consider ditching it and learning the mysql command-line syntax.

The connection can really affect the speed, especially with the PHPMyAdmin->Browser connection. Depending on the PHPMyAdmin version, it can use AJAX, or not, and so you may experience as much overhead as sending an entire PHPMyAdmin UI sans cached stuff, or still quite a bit of overhead involving the fluff surrounding the tables. While the query itself has finished, it takes a while to feed all of that data to your browser. I suggest you try to use an external client and see how that can affect it, or see whether you are using near the maximum practical throughput of your connection when you try to select thousands of rows. You can compare the used bandwidth with that when downloading a large file. With that said, I agree that PHPMyAdmin can be ditched for something more efficient.