For this sort of work, I use Adobe Audition. It handles all of the issues that you are facing, and does a great job. It is a full-featured audio-editing program, and I would not be without it. One of the best features is the removal of Transients, your clicks and pops, while keeping most of the underlying signal.
There are tons of VST's available, but I do not know how well, or how poorly, each might work with PrE. It has a couple of Audio Effects, but that toolbox is very limited. There are three things that can happen with PrE and VST's:
1.) the VST will load, and PrE will work with it
2.) the VST will load, but PrE cannot use it (PrPro will throw a one-time warning message on these)
3.) the VST will cause PrE to not load - it will hang, when trying
The freeware audio-editor, Audacity, has more power, and does allow for one to add many common VST's. There are steps on the Audacity site on setting it up to work with 3rd party VST's. Follow those directions.
This ARTICLE might be useful.
Actually it's a bit expensive at present, I'll have to wait a month or two. Are there any ways around my issues with Elements as it is?
Are the edited files available somewhere on my hard drive? What I mean is, can I access the cut up wav files directly using Audacity?
Where would they be?
There might be a way to do more in PrE, but that will depend on the VST's available, and whether PrE can work with them. There are many sites dedicated to 3rd party VST's. Some are free, and some can be very expensive. Whether they were written in a way that PrE can use them will another matter, and testing, or finding some user, who has installed each one, and used it successfully in PrE.
With Audacity, or similar, you would rip the Audio stream from the muxed AV file and work with that. I do not think that PrE has the Render/Replace function, like PrPro does - maybe PrE 8? With that function, one does the Render/Replace and what happens is that the muxed Audio stream in ripped to WAV, and that file is both added to the Project Panel, and the muxed Audio stream is Replaced on the Timeline.
With PrPro and Audition, this Render/Replace will also take place, when one Alt-clicks the Audio stream in the Timeline, Rt-clicks on it, and chooses "Edit in Audition," plus it then launches Audition and Opens that file in the program. When one just hits Save in Audition, that processed WAV file is transferred BACK to PrPro.
Unfortunately, PrE lacks that variation on Render/Replace too.
In your case, i would explore Audacity, as a stand-alone ripping and processing editor. I'd explore the various 3rd party VST's for it. Also, Steve Grisetti has done a multi-part Learning Series on Audacity, and it is on the Muvipix site. Also, a lot of folk there (in their forum, Community) use both PrE and Audacity, so they might have some helpful ideas on both programs, regarding VST's for each.