Do I need to be aware of any legal / tax issues?
Yes. You need to charge sales tax for customers in your own state (I'm assuming you're in the US?) Then you need to remit sales tax payments to your state.
Can I just start selling kits here and there, or do I need to make an LLC first?
It's probably best to start an LLC. It creates a nice separation between your personal finances and business finances and is easier to do this now than later, especially when you try to claim "business expenses" on your tax returns. It also provides a nice immunity for your own personal finances in case one of your customers tries to solder your kit, ends up burning themselves with a soldering iron, decides that it's your fault for not telling them the iron would be hot, then sues you for one million dollars. If you sold the kit from an LLC, the most they will get is the value of the LLC. If you sold it personally, they can come after your house, possessions, etc.
Should I sell them myself, and if so, use an e-commerce platform, or just do it all manually?
Big questions, and probably depends upon your kits and your expected customers, and how much you're willing to spend in time and money. If you've got some nice markup built in to your product it would be easiest to sell them through distributors. Contact some of the smaller ones as they will generally have lower markups.
You can also take advantage of a fairly new way to sell products: Kickstarter. We're testing the waters with this approach using our Arduino Drum Machine project. Some projects like the Gameduino have been wildly successful on Kickstarter.
How much work is it to support the platform? Do any of you spend significantly more time helping customers than you expected?
Build good products, write good documentation, the products support themselves 
In 3 years of business we've had exactly 1 return due to product failure. Most of the questions our support answers is on how to use our projects in the field -- not exactly what support is for but we don't mind
So to answer your question: no.
In general, is selling kits for good projects a good way to make money? Or is it more of a waste of time and money?
Don't count on getting rich quickly. Do the math: how many kits will you need to sell to make XXX dollars? Are you in it for the long haul and are going to develop a stable of products, or do you have a "killer app" product that will sell 10,000 units right away (in which case you need to be prepared to deliver on this with the logistics), or are you just going to sell a couple of kits here and there?
I'd say if you don't enjoy doing it in the first place, don't get into it for the money. If you are a steady stream of "killer app" design ideas, you could do very well
--
The Gadget Shield: accelerometer, RGB LED, IR transmit/receive, speaker, microphone, light sensor, potentiometer, pushbuttons