Yes there is slot for getting admission into state level engineering colleges through jee mains. You have to check sites of state government colleges regularly to stay updated.

You just have to fill the details and while selecting option of channel you have to simply add Jee mains as your examination.

Yes definitely you can get admission in other state engineering college through jee mains but there you will be treated as other state candidate and no benefit of home state quota will be given to you.