It's hard to say for sure, but I don't think it is likely to be offered as an option anytime soon. If you you search the forums you will find that, on most servers at DreamHost, there is still a way to run mod_php, but it is not publicized and should you avail yourself of that and your programming/site becomes a drain on your server's resources, you can expect DreamHost to ask you to refrain from using it.
I only mention this because you indicated you are learning, and I doubt than any "light" usage, as you learn the programming techniques involved, will create a problem. It is also true that your server may not have this ability at all - you can always check by trying it! If you find mod_php is available to you and decide to experiment with it, remember that it is not officially supported as a DreamHost feature, and your ability to use it could be terminated at any time. In other words, don't build a site that absolutely requires mod_php to function properly, or you may find yourself having to move it to another host should DreamHost remove your ability to access mod_php.
In addition to the information in the web panel about your PHP environment, careful study of the output of the phpinfo() function will indicate whether you are running mod_php or php-cgi.
There is no longer a "hard limit" to the number of conueries you can use on DreamHost; the only limitation is that your usage cannot negatively impact the server. There is more information about that in the Wiki, and there is also information on how you can monitor your resource usage.
If you are careful with your work, and your sites is not getting pounded by lots of visitors, I think you are unlikely to have a problem with conuery usage by your user.
As you are learning, you might consider focusing your attention on the efficient use of queries in your connections, and the judicious use of joins when necessary.
Persistent connections are not a general "magic fix" for inefficient MySQL programming - getting multiple queries efficiently from a single connection will probably produce better results in most circumstances than programming inefficiently with persistent connections.