I've not used MindJets, so my comparisons might not be so great.
Freeplane is my go to mind mapping tool.
It can do:
edit / rearrange map structure by keyboard and mouse: My mouse both, it can edit by keyboard insert and delete nodes etc. I'm not sure about moving them.
add branches with rich text and images: yes, nodes are backed with html, with a wysiwyg gui.
edit branch text in-place with adjustable word-wrap: Edit in place, Yes. I'm not sure how great its word wrap is.
collapse / expand branches: yes
auto-layout: Yes, though it isn't as good as arranging it by hand
append URLs / documents / links to branches of another map: Yes
define icon groups: It has icons, I'm not sure what a icon group is.
add notes Yes
Notable other features are:
Freeplane is a fork of Freemind. Freemind shares most of its features.
Freemind maintains a list of Mindmaping software
Outside of Freeplane/Freemind, I've used XMind, and found it to be not so great.
Your other, much worst, options are generic vector drawing tools like:
Open/Libre Office Draw, and MS-Visio, Inkscape and XFig.
I have now used Mindjets, and was very impressed.
It seems clear to me that that authors of FreeMind, (and thus FreePlane),
set out to clone the functionality of MindJets, they seem very similar.
MindJets is a much more polished product, it feels smooth.
It is also over $300, which is quite a lot. It might be worth it if I was using it on a daily basis.
It should be noted that while FreeMind/FreePlane use the samefile extension as MindJets (
.mm), they can not open each others files. There may be a common export/import format, I am unsure.