UK Citizen - Pregnant Girlfriend From Peru

Hi all,
I hope your knowledge and experience may be help myself and Carolina in our situation to better understand how we can over come some current difficulties.

I am a British Citizen and my Peruvian girlfriend is currently 3 /4 weeks pregnant. Looking at immigration laws i have found a few areas that apply to us that if anyone has experience of knowledge of the below it will hugely help us for the process moving forward. (currently this is scaring the shit out of us!)

1. applying for initial Tourist Visa to the uk for Carolina...
when researching this i have seen much information saying that it would be best for me to sponsor her visa. she needs to have sufficient funds to support herself for the duration of the stay and that it is not obvious that she will look to remain in the UK past her visa period or work during the time.
- my worry is that if i sponsor her and she turns up at the immigration office with a BF sponsored visa and a bump on her belly they will see this as a way to sneak her in the country to have the child and refuse her entry.

2. once in the uk, if said child is born there but is a short time after the pregnancy; what issues may we face with Carolina and the child being potentially kicked out of the country with a new born. (i know that the child would be British by default but no-one wants to separate a mother and their new born child - and she is not currently have the support system of family in Peru for me to be happy to let her go home until paperwork is sorted) or will Carolina be the only person who is legally obliged to leave? meaning either myself or Carolina being away from our newborn child for a period of time until paperwork can be processed?
an additional worry is that she as a mother but not of a British decent will not have any rights in the eyes of the British immigration office unless we are married.

3. is there any way that visas can be accommodated for Carolina if we are not married prior? - we have been together 4 months and this has been a huge shock to us. for being such a new couple we would like to try and do this without Marriage and save this joyous occasion for when we are able and stable enough to do it correctly as she has always wished.

4. Lastly, it is hard to see and find out if we were to come to the uk before with Carolina on a tourist visa if she would be entitled to free NHS pre natal care as being the child of a UK citizen. and if not its rough costs if needing to pay for scans etc.

Any help here from experience would be greatly appreciated.
Many thanks
Daniel and Carolina