Mortgage rates differ from place to place, and as per market rates, the rates vary in one or the other way. So, if you like the deal for your home first talk to an expert and then choose the best deal which you like.
Equity is the deemed value, minus it from the amount you still owe on the plan.
Under equity, if you got more equity savings then there are more chances of getting a better interest rate which and on the other hand, if you take equity then there are higher chances of lower monthly payments.
Refinancing means taking a loan from another company on different terms.
The main reasons for refinancing are if the person wants to switch to the lower payment options, for shortening your term.