Can I use ExtJs without License like JQuery? Can I publish website using ExtJS without any permission from them?
No, you would need a commercial license unless you are releasing the source code of your application with a license compatible with GPLv3. (ExtJS - Licensing Overview)
Currently a single developer license which allows you to deploy Ext JS on unlimited domains and sub-domains on unlimited servers costs $329. (ExtJS - Online Store)
ExtJS uses a Dual License business model for its software development libraries. Basically you are required to contribute either by purchasing a commercial license, or by placing your application under an open source license compatible with GPLv3. By purchasing a commercial license, you will no longer be obligated to publish your source code.
ExtJS is available under both GPL v3 for open source projects and under a commercial license for proprietary applications. The license terms for ExtJS can be found here; have a look at that page for details.
That said, I'm not a lawyer, and if you're unsure about what's ok in your particular situation, you should seek legal advice from a professional.