You should not have to worry about the cord.
The pups come out in a sack and the mother chews this off and chews the cord and eats the placenta. I did not have to do anything except watch.
Straining is that she will look like she is trying to have a poo, pushing but nothing is happening. If this happens for two hours she needs a vet as a pup can be caught.
Again all went well here.
I will keep an eye on this thread in case you need more advice.
It is great that you are desexing after this.
Make sure your bitch has a good diet, puppy kibble, maybe some raw egg with the shell crushed in it, sardines, raw meat, yogurt is good too.