I had the same problem recently. The rats will generally try to move in with the rabbits and eat their food, wee and poo in their space and keep out of your sight all the while. Whilst they tolerate each other the bunnies certainly won't like it. Mine didn't. There is a risk a dominant male rat will attack a rabbit, a nasty bite can lead to infection. Rat urine also carries a lot of diseases which can affect us humans and they might be able to pass on VHD. They can certainly pass on other parasites like mites and lice. I'm pretty certain my Emrys who suffered from ear mites this year caught them from a rat.
The general rule of thumb is that if a rat can fit the width of his head through mesh his body can follow. They can collapse their skeletons and get into all sorts of places. They can also chew almost anything, breeze block for one thing and wood is a doddle I'm afraid. Perhaps some metal panels around the edging of your shed will make a difference. You have a problem if rats have settled in your garden. One male will establish a 'deem' and install a few females there. And of course he will breed with them. Rats breed in the same way as rabbits, they ovulate spontaneously and gestation lasts around three weeks. It is estimated one female rat can produce 200 offspring in one year. They are also very wily and clever. They know how to stay hidden and instinctively know to avoid most traps. Your best bet is to kill them I'm afraid as they will only breed and breed until you are over run and then the council will impose and order on you to pay for pest control which can cost a fortune. I would steer clear from poison which causes a cruel death and cannot be controlled meaning other animals could suffer. But traditional rat traps, fenn traps or if you know someone with an air gun go down that route. It’s not nice but humans have died from rat bites, it’s not something you take lightly.
We live near a river close to the Water of Leith spring and as such are very much in the country so rats are a problem here but they have only become worse since the council reduced bin collections, go figure eh?
The rats you have seen in daylight will most likely have been inexperienced juveniles. The true adult rat is a master at travelling unobserved. Juveniles will mean there might be a nest nearby. My advice would be the following:
Don't let your child/children into the garden for the time being.
Bring the rabbits indoors.
Examine any rabbit bedding with a prodder/stick and wear latex gloves (coming into contact with rat urine can lead to leptospirosis).
Most of all you need to nip it in the bud. Humane traps rarely work and you would need several to cope with the breeding rates. A quick clean death is the most efficient way I'm afraid.
I hope you manage to find a solution. As I said I have had a lot of experience with this so if you need any more advice just let me know :wave: