Yes, you can usually borrow more by remortgaging. When you apply for a remortgage, you can apply for a larger mortgage loan. Whether you'll be approved for this additional borrowing depends on whether you pass the lender's affordability checks, as they'll need to ensure you can afford to borrow more.
Yes, you can remortgage to consolidate debt. By release equity from your home by remortgaging, you could pay off your debt. However, you'll need to pass the lender's affordability and credit checks to be approved for a remortgage. If you have a poor credit score or expensive financial commitments, you might have access to a smaller range of lenders and may be offered higher rates, which can make it more difficult to remortgage.
Yes, it is possible to remortgage and borrow more with a shared ownership property. To do this, you are likely going to need to switch to a mortgage lender who is willing to offer you a larger loan. It can be difficult to know which lenders are likely to accept your application. Working with a specialist like a mortgage broker can help you find the best lenders to apply to.