That's just an ordinary brown rat - the saying is you are never more than 6 feet away from a rat, so they're everywhere. Don't leave anything food related in the garden, including used barbecue coals, which will have fat and juices from the food you cooked on them - it's unlikely to be the rat digging in the lawn, unless there were bits of food in there too, or the rat/s have a burrow beneath the lawn. If you see rats most of the time, you may need to call a rat catcher or pest control company - they usually work out where they're coming from.
In respect of the lawn, you need to get a lawnmower asap if you haven't got one - get a Flymo or other make hover mower. You won't have stripes in the lawn,but you will be able to cut it when it's a bit longer than it should be. Right now, though, it's very important to get rid of the length - use the strimmer by all means, but just get it down to around 2-6 inches, then run the mower over it. Alternatively, cut it with shears - it's hard work and backbreaking, but will do the job. You will not be able to get the strimmer to cut it evenly, whatever you do,but don't wait any longer to cut it - once the weather turns and autumn rains arrive regularly, you'll be in trouble if its still long. Once you've mowed, you should be able to inspect more closely how much damage there is and quite what it looks like, and also more clearly see what might be causing the damage - it's easier to see any creatures out there plus any droppings if the lawn is 2-3 inches long. Foxes often dig holes in loose soil or lawns, especially new lawns or those on light soil.
You will probably find the grass looks brown once it's been cut, possibly even yellow in places - that's because its been left to get far too long, and it should green up over a week or three. Keep the lawn mown ongoing, whenever it needs it, unless the ground is waterlogged or frozen/frosted.