How can you increase the block time from 10 minutes to 40 without a hard fork?
Just by producing blocks every 40 minutes. If any difficulty reduction will be needed, then soft-fork is more than enough to cover it. But I think moving coins off-chain will be enough to decrease on-chain fees, and to decrease on-chain rewards, which will encourage miners to turn off their machines.
How is Proof-of-Stake implemented exactly?
By making signatures. Instead of signing a regular transaction, you sign something else. For example, in signet, if you want to sign a block header, you make two transactions, there is "to_spend" and "to_sign" transaction, you sign this "to_sign" part, and place your signature in the coinbase transaction.
If the altcoin is inflated, how can you return to the main chain that is not Proof-of-Stake based?
I thought about 1:1 peg (of course 1:1000 or other values are possible, but then it would mean just moving the comma, so for example using millisatoshis, instead of satoshis, as in LN).
If there's a direct correlation between bitcoin and the altcoin, how isn't it completely dependent on Proof-of-Work?
As every sidechain, it has to reach "a peg", not "the peg". As long as all coins are covered by the chain they are pegged into, it works.
Make a thread, explain your mechanism in a clear and detailing way.
It is in progress, I just decided to mention that it is ongoing, but I could patiently and silently work on it without informing anyone, and then suddenly reveal everything at once. So far, I shared parts, where there is very little chance that someone will destroy that easily (also, Core developers already discussed some parts of it, for example "temporary forks", so they are aware of what is possible, they may not be aware only about that "no fork required" part, but I think they should be, because this was also posted (not by me) to the mailing list). But there are other parts, that needs to be hardened, they are too fragile to be shared with Proof of Work enthusiasts, because then they will abuse the system by making chains, and pretending that "this idea was tested, and it failed", which is not the case. In the same way NameCoin was presented as a BitDNS implementation, while it is clearly not the case, as it lacks some features (described by Satoshi).
I'm sure there will be interest if you've thought of something that's significantly utilized somehow.
I can share unstoppable ideas, that are hard to destroy, and I already shared that, the only thing left is answering some questions, and clarifying, what is possible. But still, I don't want to share more fragile parts, because I don't want to repeat NameCoin's mistakes.