Yes, we do WordPress maintenance for our client's websites. The focus is on WordPress security. We perform critical software updates (WordPress + Plugins) by our development team on a monthly basis in a staging environment (a copy of the live website). Once tested for theme compatibility we push those updates to production (live site).
Yes, all the websites we build are responsive in nature. We go the extra step by customizing the mobile experience - prioritizing above the fold content to 1-2 focused areas and improve the call to action placement, making sure it's not buried at the bottom.
We build custom WordPress themes. If you have a limited budget of around $10,000 we do utilize a pre-built theme to accommodate budget constraints. See our in-depth article on The Difference Between Custom WordPress Theme vs. Template.