I just noticed: In your video, the power supply is set to 7.8V.
The chip needs something between 8 and 28 V to work properly.
So you're simply operating out of the specifications of that IC.
Increase the voltage to at least 10V. Use proper (read: much thicker than what you're currently using) wires for power supply.
Again, the IC is positively ancient. It doesn't have any undervoltage detection, so it's no surprise that
When I start about @ 2.3V, the sound is somewhat good, but very quiet. When I bump it up to about 4V, the sound it totally distorted.
since you're practically pushing the sound input through insufficiently biased transistors.
When I rise it again to about 7V, the sound is really good, but one of the speakers starts to do that tick tick tick...
aligns very nicely with that.
When I connect both speakers to one channel, even at bigger volumes, it goes just fine.
Yeah, then you reduce the load impedance, and thus, the voltages that the transistors can supply get lower, and your sound gets compressed to a smaller voltage range. I'd claim "it goes just fine" is not really true. It might sound like you hit the "loudness" button on an 70's stereo.
I'm still recommending a not-as-obsolete amplifier ICs. There's literally thousands out there which aren't as terrible.
I do personnally know the seller and he sells a lot of these every day, so he has to buy new one like every week or so.
It's nice that you know your seller personally, but he's selling you ICs that are marked as "obsolete" at digikey, mouser and other larger retailers, and that the original manufacturer (ST) doesn't even list anymore. In other words: he's selling you the crap that no-one but beginners, who don't know better, buy, or people that have to repair ancient amplifiers. Even if DIY audio enthusiast often swear by some older amplifier designs, this IC is simply not worth using in the year 2017. There's nothing comparably good about it.
Go to the website of the former manufacturer of the TDA2009A (it's http://ST.com), click through the audio amplifier ICs and select a Class AB amplifier (modern class AB will sound a lot better than class B, and be much more efficient than class A, I actually don't find class B in the wild very often anymore; they're basically bad amplifiers, to be honest) and build the schematic from the datasheet. You can often get them for free as samples, or for little money at online retailers.
Don't just buy anything your local seller recommends. He probably has a stock or an extremely cheap source of obsolete parts and is trying to make money with inferior products. That is bad style.