Effective labour depends on three things working harmoniously together- the powers (ie. contractions) the passenger (the baby- size, weight, position of baby's body and head) and the passages (your pelvis mainly)- any one of these things on their own, or several combined, can lead to delays or complications in labour.
Everybody is different and as much as doctors would have us believe, not everyone's body works in the same way, particularly when it comes to labour and birth.
It is difficult to speculate, as we weren't present for your birth and don't have access to your labour records, but we do know that progressing quickly in labour can cause heart rate changes in the baby- this can happen particularly in women who have had babies before, as the baby descends quickly on to the pelvic floor. Compression of the fetal head during labour and birth can cause the heart-rate to drop- this has a phsiological purpose in that the baby releases adrenaline in response and the baby is often therefore more awake post-birth, ready to bond with the mother and breastfeed. Well-grown, full term babies often have no problem coping with this descent and can quickly recover their heartbeat, however some babies, in the absence of other factors, don't cope very well and that is when the birth ought to be expedited- a low heart rate for longer than 10 minutes (in a well grown, term baby) means the baby can suffer from oxygen deprivation. When the heart rate has been down for 3 minutes, this is classed as a bradycardia, and preparation for birth, in whichever method, needs to be made.
Caesarean sections can happen when a mother is fully dilated. They can even (though rarely) happen when the baby's head has already been born. If the cervix isn't completely open, the baby cannot descend, even though, particularly in women who have had babies before, cervixes tend to open faster.
As said above every woman is different so pain relief that works for one person won't work for another. There is also not set or defined rate for cervial dilatation (though many hospitals 'prefer' you to dilate 1cm from 4cm onwards an hour and say you're defective if you don't)- having a vaginal examination only gives you what dilatation you are at that set moment in time- you can be fully dilated 20 minutes later, or 2 weeks later- labour and birth is very unpredictable, particularly in women having subsequent babies (which could explain the quick nature of your birth and why the decision was made for a forceps and not a caesarean section).
I am sorry that you did not have the best experience, and hope you are both recovering well. Please don't feel as if you have to be 'greateful' or any other expected emotion regarding your experience- your emotions are your own and the way you feel about your experience is always valid. What happened would be regarded as a traumatic experience by anyone, professional or mother alike, but it is important that you process it as you see fit.
I highly recommend a birth 'debrief'. Many hospitals offer this as a specific service but some do not- either way you are entitled to one- you could ring the main labour ward or contact the head of midwifery- they will always point you in the right direction. The service will offer you an opportunity to ask further questions, see your labour notes and process what happened, though you may not feel ready for this for several weeks, months, or even years.
Sending you lots of love- and ofcourse many congratulations!! xx