Speaker Questions for the Audiophiles- are these flat wall speakers a good idea for a cheap system?

I've been a big fan of the "Tech Ingredients" youtube channel after I watched their video on their air conditioner that used desiccants instead of refrigerants, and this video of theirs peaked my interest greatly- I don't have tons of money with my new house, but I DO still have all of my dad's old audio equipment circa ~1985. Do they seem like a good, cheap DIY speaker project (especially on woodworking skills that have atrophied over the years)? Or should I do something like the Sunflowers instead? They would be going into a closed Den with one "enterance" and a landing/family room area above it, both largely enclosed with rug/carpeted floors.

Link to sunflowers: https://sites.google.com/site/undefinition/diy-sunflowers

In a word, Yes!

I built a pair of these a few years back based on info in the Parts Express catalog.  In fact, I use them everyday.  I'm listening to them now.  Very airy, open.  Beautiful with acoustic music.  Mine are fairly weak down low, but a sub would fix that.

Edit:  I answered only half of your question, I suppose.  The sunflowers look good, but will run a good bit more money.  Probably not lacking in the bass department, either.

Fascinating video, I'm going to have to play with the panels for the garage stereo.  I'm skeptical but there's almost nothing to lose but a few hours of time dinking with cheap parts.  (sort of like my Miata...)

I'm a veteran audio DIYer and have built a lot of speakers, both my designs and kits.  DIY speaker building takes a large investment in time and money, and unless you're using an established reference design, you'll probably be disappointed.  Craigslist can be cheap and near instant gratification.  It's possible to endlessly tweak a design into submission, but for me these days I'll just borrow what somebody else has proven to work.  The Sunflowers look like a nice design and he mentions Sigfried Linkwitz, one of the bright stars of audio speaker design who used dipole designs with great success. 

The problem with building something like this is that you won't know how a speaker sounds until you listen to them and it's  a large investment of time and money to put them together, so there's a risk there.  Room compatibility shouldn't be overlooked either.  The Sunflowers probably rate a bunch higher in WAF (wife acceptance factor) than hanging random panels in front of a wall.  If you build either, I'd like to see you update the thread.

If you're into unconventional, you might also consider Linkwitz' Minis, which sound damned good.

https://www.linkwitzlab.com/LXmini/Design.htm

There are others here who have experience on this matter, but they haven't checked in as yet.  Must be busy.

I have no such issues.  Here is a link to the P-E info:

http://projectgallery.parts-express.com/speaker-projects/dml-flat-pannel/

I have watched the entire internet about flat panel speakers.  Ok, maybe not the entire internet.

Consensus seems to be that they sound great, but the frequency response is reliant on the qualities of the panel and how they sympathetically vibrate in the overtone series given the relatively flat response generated by the driver.  This means that frequencies on the 3rd and 5th overtone get boosted while others get attenuated.  In truth, it seems to be a fantastic way of getting cool, DIY sound as long as you're in it for the neat packaging and cheap, homebrew approach, and not the audiophile grade sound.

Think of it this way:  You're taking remarkably cheap speakers and using the substrate to maximize the vibrations coming from it much like Bose takes a 3" driver and puts it in a folded horn to make it a subwoofer.  You're making the most of a cheap driver as opposed to using a quality driver.  It's a great exercise and makes way more sound than the $15 investment suggests which is its calling card, but you're still limited by the shortcomings of the driver.

Low frequencies will suck.  You could use a larger driver to be able to excite at those frequencies, but my guess is that the panel would have to be 20' long to effectively generate anything below about 180hz, especially with the rigid materials you would need for the higher frequencies.

Also, I have read that they really need to be located a significant distance from the wall behind them.  In one of those videos where the nerdy guy (not putting him down... I'm nerdy too) tests a whole truckload of materials and sizes, you'll notice he has his panels about 3' away from the wall.  My guess is that the pressure waves from the back of the panel reflect back and counteract the vibrations (standing waves) at certain frequencies if they're too close to the wall.  Moving them away reduces that effect.

Having said all of that, I think I will try some.  I have a big projection screen for outside viewing.  I normally hang it on the garage but I want to put two fence post pockets in concrete in the back yard so I can stab two poles in it to stretch the screen across.  I also have a ton of pink insulation board and some 5.25" marine speakers.  I'm thinking about making flat speaker panels to hang on the outside of the posts.

When I saw that video I got all excited about it. Then I got less excited about it, for no real reason. Maybe just a bad taste in my mouth from the clickbait video title. "Worlds best speakers". Really? No not really. Good? Sure. Cheap? Very. Worlds best? Just stop. 
I may be wrong, they may be the best, but I think if the design was inherently better we'd see a whole bunch of outfits selling them at hugely inflated prices. The market seems to think it's not worth pursuing, so I think I'll stick to traditionally designed speakers and proven box designs. 
I am very interested to follow along if you do decide to give it a go. So for that reason I say do it. 

I think anyone who makes a pair of the flat panel DML's will be genuinely impressed with the sound.  Also, your friends may be amazed that those pieces of styrofoam on your wall are making all that sound.  I'm tempted to make another pair with  2' x 4' ceiling tiles and see if it's any better at reproducing low frequencies.

87 pages of discussion on the subject of flat panel speakers

Well engineered flat panel/planar speakers do sound pretty good - Magnaplanars, for instance.  Somehow Magnapan never quite made the jump from mylar to ceiling tiles though.