Honey
Your ESA(CB) claim is in your name only, if your partner claims CA then that will be in their name only so there is no issue of overlapping benefits.
If you are receiving what is now called legacy and not New Style ESA then providing you are eligible then Income Related ESA can be added to your claim.
You cannot make a new claim for Housing Benefit unless you meet very specific criteria and your current circumstances and planned ones prevent you from meeting this unless your partner has an enabling award of DLA or PIP.
Council Tax benefits are not part of UC, if you meet your Local Authorities criteria then you should already be receiving a discount.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
