There are several ways to upload images to be used in the content. The first and most common is through the rich text editor (RTE). However, I see you're using an outside application to generate your HTML content. Please find my answers to your questions below:
1. Is there a way to establish a default for a specific group such that images stored in that group have a different file structure and/or folder naming convention than the default?
Unfortunately, this is not available, so there's no way to change the way images are stored in a particular place (or sitewide).
2. Secondly, is there a way to upload an entire folder of images, rather than uploading images one. at. a. time. for. so. so. so. many. images.?
If you're using a hosted or on-premise environment, you might be able to workaround this by utilizing a theme to upload images in bulk. This is covered as outlined here:
On a smaller scale, you might use the "Manage Files" button in HTML and Formatted Text widgets to upload static images as outlined in the comments here:
Unfortunately, this is not available for Jive Cloud.
3. Thirdly, are there tiles that will allow me to enter whatever HTML I like, or is that only possible in the widgets on the (one per group ) Overview Page?
For security and performance, not all HTML will be supported and will be filtered by the macros and filters mentioned. If you need to add custom HTML, the best place will be via an HTML widget as outlined in the knowledge base document from question 2.
I hope this helps. Let me know if you had any other questions.