Hi there.
Are you the poster from Norfolk? If you are, as we discussed before you really shouldn't be using an automatic trap in a situation with more than one cat, as it will snap shut in front of the other cats there and it will be very difficult to catch more than one. However at the time we last messaged I rang round all the rescues in Norfolk and was horrified to learn that not one does proper trap/neuter/return. There are no trappers that go out and no one has manual traps. So we are stuck with the situation as it is and we will just have to hope for the best.
So to answer your questions. It's absolutely fine to catch one/both the night before. The second the trap shuts cover it with a sheet. You want to avoid the others seeing what's going on. Then bring it into the house. You will just have to leave it in your kitchen/hallway/wherever overnight. Don't leave it outside as, apart from being cold and unpleasant for the cat/s inside, the others will witness the distress of the cat inside the trap for hours and they will never, never go in it themselves.
Once trapped DO NOT OPEN IT! Talking about putting food and water in makes me think you plan to open it to do this. You mustn't put food in anyway - they should be nil by mouth before the op. They could have water, but as you won't be able to do that without opening the trap then they will have to go without. It won't be nice for them but needs must.
A thing to bear in mind about an automatic trap. The end normally slides up, where you put the food in. When you set the trap this must be cable tied closed as cats, particularly distressed ones, can push the end up.
Then what's going to happen to them after the op? Will the vets keep them in for a few days? Ideally they should be kept inside for around 3 days, but certainly for a minimum of 24 hours after the op. How will this happen?
So the mum and small kittens. Well they can't leave the mum for a while. So unless you can arrange somewhere for the mum to go with the kittens then you will need to wait until the kittens are around 4 weeks old. At that stage trap the mum and get her done. While she;s being done bring the kittens inside. Start handling them from 3 weeks to get them and and mum used to the idea and to make sure they will be pick up-able. They can then join her wherever she is recuperating for 24 hours. If you can't find anywhere to take them then return them all to the greenhouse but make sure you interact with them a lot, every day. By 8 weeks, as long as you've done enough taming, you should be able to find them homes one way or another. I'm sure CP or the other rescues will be able to help with cute, friend 8 week old kittens.
Thank you for caring and for doing this and I wish there was a proper rescue around to help you.
And if I've got it wrong and you're not the lady from Norfolk then let me know where you are and I'll try to find proper rescues where you live.