Obviously she will have to prove to the ECO (entrance clearance officer) that she has ties back home which outweighs her reasons not to overstay in the UK.

As you have said if the visa is granted then yes she would have to pay for the birth of your child.

Also if the visa is granted then your child would acquire british citizenship as you yourself are british.

Please don't think this is easy as it looks because it is rather difficult to obtain this type of visa even at a small outlay of just £80, the intention to return home for her will be looked at scrupulously by the ECO.

Please search here on this subject as it will give you all the info you need regarding the process and I am sure others will provide further advice regarding this.