So this year I found out that when compiling a java applet in Eclipse Oxygen that java applets are about to be obsolete. Is Oracle going to release a new API that allows applets in websites? If not, what are the alternatives and what are their advantages and disadvantages?
Is Oracle going to release a new API that allows applets in websites?
No. They are accepting the decision of browser manufacturers to remove all plug-ins from their rendering of web pages. See Moving to a Plugin-Free Web for more info.
If not, what are the alternatives ..
HTML 5 and JS can replace the vast majority of the reasons people ever used applets.
.. and what are their advantages and disadvantages?
The primary advantage of JS / HTML is that - it is supported in modern browsers. It also helps keep the web page a keyboard navigable thin client.