Could some form of DH be used for the secure channel and passive receiving? Like, I publish some pubkey (A=a*G) of mine, you generate some random one-time private key (R=r*G), and publish the (R, H(r*A), unbalanced transaction encrypted with H(r*A)) to the blockchain. I can claim this TX at any time (unless you double-spend?), after which it could be purged from the blockchain.

I have no idea, sorry. Hints could be in the code of Electrum, but I'm no coder so can't really dig too deep there. I suspect it's simply implemented that way, that it's calculated only for new wallets.

That's right. All I'm saying is that entropy after hashing can't be higher than before hashing. Cryptographic entropy is a measure of randomness, by definition. In physics, it is something else, though. How could 12 word seed not be possible? It's simply a way to encode 128bits of data. For more data, you need either a bigger dictionary or more words.

Don't confuse entropy with length. Yes you can feed anything to sha256, but if I feed it "aaaaaa" and get some 256bit number, it won't really have high entropy. It may LOOK like it has, but it won't. Somebody somewhere could get the idea to do the same what you did and take your coins. The source should be as unpredictable as possible. You could use dice rolls to generate your key - you'd need about 50 rolls for 128bits of entropy though. As for the 13th word, I guess when importing directly from user generated seed, it doesn't calculate it. Would need to investigate this a bit more, though.

Honestly, I have no idea as I don't really follow any of those coins, sorry. I take it it's easier to be profitable with PoS, as you don't have to spend your resources (electricity).

128bit = 32digit -> 12 words, 256bit = 64digit -> 24 words. Apparently, 128bits is considered more than enough considering number of possible combinations and time it would take to brute-force them all, but you're free to use 256bit. I wouldn't agree that your wallpaper is a good source of entropy, though. Taking a photo in low-light could be, considering all the noise introduced when doing so.