Am I eligible to contribute to a SEP IRA for my 1099 income? Do I have to have a business (sole prop, etc.) established to do so?

Yes, you are eligible to contribute to a SEP IRA on your 1099-MISC income as this income is self-employed income. You are self-employed and are a sole proprietor. You do not need to have a legal entity such as an LLC.

Please read this IRS document for more information.

I am retired and have a LLC which generated income of 1099-MICS, am I eligible to contribute to traditional IRA account?

Yes, you can, as long as the 1099-MISC reports self-employment income for services you provided as a member of an LLC that carries on a trade or business.