When you use a free webmail client from any provider, you're largely at their mercy, so if you're looking for a better user experience you may need to look for real mail client (product recommendations are off topic for here, but outlook and thunderbird are both popular). However, regardless of what client you pick you want to connect via IMAP and not POP3 to keep your mail safe on the remote servers.
See: IMAP vs. POP3 vs. EXCHANGE
So to answer your three questions in reverse...
Can I go back to previous version of Yahoo Mail?
Most likely not.
Can I use any client which can give me better organization of my emails?
While Yes in theory, you're safer sticking to an IMAP compatible client. Thunderbird is free, Outlook costs money. Either should provide an exponentially better experience than any webmail client could.
Can I forward my mails to any other email address? If I do that, can I still send emails using my Yahoo email?
Yes, sort of - this is easily done with a client where one program would check both, but send from one by default.
See Also:
http://www.ghacks.net/2013/10/10/access-yahoo-mail-using-pop3-imap/