Taking a partly developed foetus out of the mother? This is currently not an available method.
However, there are methods out there for the problems you described:
One method allowed in the UK is called a "three-parent baby" and the following guardian article explains it fairly well:
The procedure replaces a small amount of faulty DNA in a mother’s egg with healthy DNA from a second woman, so that the baby would inherit genes from two mothers and one father. The idea is to prevent certain genetic diseases being passed on to children.
[Image Credit: BBC Article] Click Images for full resolution!
Host (or gestational) surrogacy is when IVF (in-vitro fertilisation) is used, either with the eggs of the intended mother, or with donor eggs. The surrogate therefore does not use her own eggs, and is genetically unrelated to the baby. It is physically more complicated and considerably more expensive than straight surrogacy, (although many IPs can have some costs covered by the NHS) and always takes place in a fertility clinic.