I'm sorry to hear how much time you have spent on this. The fact that you were willing to post instead of just giving up completely is a good sign.
It is true that the current theme engine is a bit difficult to figure out if you don't know PHP since it uses constructs besides simple variable assignment to HTML. If you look at the main Coppermine page under "what does the future hold", you'll see one of the items is "template-building tutorial". The theme upgrade guide could be improved and your post helps to figure out how by giving us a story of someone who doesn't know PHP, but should be able to create or modify a theme based on your other computer skills.
I just joined the dev team and didn't create any themes in 1.3 so I don't know the differences first-hand. I do know the 1.4 theme system fairly well and while I agree it is more complicated, it is fairly well organized and the features gained with the new system are worth the effort. The themes board has some basic tips that supplement the docs and has many threads with people learning how to customize or upgrade their themes. I would suggest to go step-by-step and see what issues you're seeing. If you've read the docs and searched the board with still no success, feel free to post on the themes board for assistance. For each separate question you have, please use separate threads. If you have upgrade-related questions unrelated to the theme, please post here. Otherwise, theme questions, even if you are upgrading, should go on the themes board.
A quick answer about the new system buttons - no they are not necessary. You should be able to use your old HTML in the $template_sys_menu and $template_sub_menu variables. (For follow-up questions about this, please see the themes board and post there.)
I don't know if it's critically important for you to upgrade to 1.4 (someone else might know better), but I would recommend it. Yes I realize you ran into a lot of issues and spent a lot of time already on it. But the support is here and I think most of your issues are theme-related. (Oh, the new config page is admin.php.) You could verify that all your problems are theme-related by using the classic theme in 1.4 and make sure things work. Then continue to work on your theme upgrade.
If you do want some paid help, please post on the freelancer board. But I think that it should not be necessary, although I understand if you just want to be done with it and have someone else do the upgrade for you.
There are a number of features that make 1.4 worthwhile, particularly the plugin system which means you can add-on to your Coppermine without modifying the core code. The new theme system does allow more functionality as well, once you get used to it (and maybe more usefully once the template-building tutorial is written).