Yes, eligibility for COBRA does not affect your eligibility for Medicaid or vice versa.  Medicaid also offers premium-free coverage (in some states Medicaid charges a nominal monthly premium for some adults) with no or nominal cost sharing.  As you consider options, you will want to compare provider networks, cost sharing, covered benefits, other plan features.

Yes, can choose.  Eligibility to elect COBRA does not change eligibility for marketplace subsidies, or vice versa.  If you are receiving UI benefits, you will also be eligible to enroll in a zero-premium marketplace plan with comprehensive cost sharing subsidies.

You can continue on COBRA unsubsidized until your 18 months of COBRA eligibility ends. In addition, you will have the option to enroll in marketplace coverage when the subsidy ends in September.  Loss of the COBRA subsidy will make you eligible for a special enrollment period (SEP) to sign up for marketplace coverage.

No.  People lose eligibility for the COBRA premium subsidy when they become eligible for other job-based health coverage.  You are required to notify your former employer or COBRA administrator if you become eligible for other job-based health coverage.  If you do not, you might owe a penalty.