So will you be keeping this little bunny?
Maybe .... just until a rescue spot comes up .....
No. Rabbits don't need dust baths. When you say the fur is matted, does it seem wet? Or just tangled? If it's wet then it can be indicative of dental problems, so would need a vet check-up.
The fur is not wet - it is just a mass of tiny dry matts
He can go out in the garden every day (unless it's bad weather) if he lives inside, except in winter. When the temperature difference is high - i.e putting him out of a heated room and into a windy garden - that's not ideal conditions for him to go out into, as it can cause health problems. But when the temperature inside is the same as outside, or even if it's slightly colder in than out, then he can go outside. You can buy runs online for decent prices - it needs to be a good size, though. Definitely not one of those triangular ones, they're teeny tiny. There'll be some good runs on here if you look at various housing examples or just search the forum. Obviously, by all means, let your father build one, but sometimes it's just easier to buy one! The main thing it is big, and NOT using chicken-wire. Foxes get through that in seconds and it's not durable at all. You want proper mesh.
My dad is pretty good - he built the rabbit runs when my mother and I used to show - it was many years ago - but I'm sure it will all come back to him 
Living outside isn't necessarily healthier than living in. There are various factors you'll need to take into account, however. For example: predators. Are there foxes and cats around you? If so, then it'd be safer having him indoors. Company is another factor - rabbits ideally need at least one other rabbit companion whether they're living in or out, but it's more important when they're outside as obviously being indoors means they get your company!
We have lots of cats and foxes around - indoors it is then!
My rabbit lives indoors, but he used to live outside when he was bonded (he's single now). I much prefer him being inside, as I don't have to worry about him so much! And going outside in the cold, muddy and rainy garden was not pleasant in winter
If you're going to be keeping him indoors for more than a few days, a dog crate is too small really. Don't go for one of those pet-shop plastic cages: they're absolutely tiny and way, way below the minimum for rabbits. I personally keep my bunny in a dog crate with a puppy pen attached. The crate door is permanently open and he gets to roam around the room in the day, shut in the pen/crate at night.
He comes out whenever I am home and has the run of the room I am in
What are you feeding bunny? Have you checked whether it's a male or female? If you're keeping him, he'll need vaccinating and neutering if he's not already done.
I am feeding him a rabbit mix, free access to meadow hay and have given him straw as bedding. Is that ok? He has had a little bit of carrot - but I don't want to give him too much as I don't know what he has been fed before. Have also tried a little bit of apple - he not so keen on that! Am going shopping this afternoon - broccoli? parsnips? Cabbage leaves? cauliflower? I'm currently not working - due to start again Monday so will get him to the vet end of next week to have him checked and vaccinated- just myxamatosis? ( sp) I have not yet checked his/ her sex but will do that this week. I didn't want to terrify him!
Again, if you are going to be keeping him, I'd STRONGLY recommend getting him insured ASAP. Insurers generally have a two week exclusion period at the start of the policy where you cannot claim for anything, but other than that they're a life-saver for me - literally. I'd get it done sooner rather than later so if any problems crop up within the next few weeks you have a chance of claiming for them!
Again something for the first weeks wages!
