Is there a way where I can distribute a public decryption key, but the encryption key cannot be computed from that key, so that anyone who reads one of my encrypted message, he/she can be quite sure that it was originally written by me?

So basically it is the traditional asymmetric encryption-decryption scheme, but in reverse.

If you want to prove to others that you took the screenshot on a specific date and not later, you will not be able to do it yourself, you will have to rely on some common trusted third party.

For low importance issues, this can be accomplished by simply posting the image on some well known public service where the date when the image has been posted will be mentioned. Be sure to check beforehand that this service does not offer any possibility to modify the picture without altering the upload date!

For higher importance issues, you will have to contact a bailiff or a notary. By being present during the screenshot, they will be in measure to vouch the date and the conditions when it has been taken. For instance I've read about such procedure being used when someone wants to keep a proof of the existence of a security flaw still valid even once the flaw itself has been corrected. However, if you go that way I would strongly recommend to check the Law StackExchange website before engaging yourself into anything.
