Oh gawd, now I have to think!
For a start, a yard as opposed to a garden is generally no big deal providing you aren't after a big dog/very active one and are able to walk him daily. That its secure IS important, so no problems there.
A good homechecker will want to know whether you all want a dog, who will be the main carer, how often/where he will be left, what you will do if you go away (and any forthcoming plans to do so), what you think of dogs on sofas/beds (to disagree is not an automatic failure but it will rule in or out some choices of dogs in their rescue), how old your DC are and how you feel about them interacting with the dog (ie will they be allowed to pull him about, cos yes, we do meet that type!), the size of your home (although they won't expect a tour of bedrooms, honest!).
Rest assured that the homechecker isn't there to judge because you've got the washing up or ironing on the side waiting to be done - this isn't about your home per se but your attitude towards one of their dogs in it and whether he will be safe and loved. They will also probably ask about any other pets and regular small human visitors and want proof that the house is yours or that your landlord, if you have one, allows pets.
You'll doubtless be told that the rescue will require written assurance that the dog will be neutered if this is for some reason not already done (eg if he is too young yet), and to confirm that you will return him to them if ever you can no longer keep him.
Depending on the rescue they may suggest/introduce you to selected dogs which they think might suit as the one I volunteer for does or they may just invite you to take a wander and see which appeals then advise if he/they might suit. I would expect them then to homecheck and to invite you back to spend time getting to know your chosen dog, taking him for walks etc until both you and the rescue are happy.
This doesn't always happen, some rescues will just homecheck and then say there you go - "my" rescue often has families visiting a couple of afternoons a week over the course of 3 or 4 weeks, to allow the family and dog to get to know each other and satisfy themselves that the two are compatible.
And a bit of advice?
Go with an open mind. When I lost to cancer my small collie cross girl, whom I'd adopted as an 8 year old, I went looking for a similar small girl but a bit younger. In time I found my ideal dog - a 9 year old, huge, long haired German Shepherd boy... and he was the best thing which has ever happened to me!
You too might have one idea but surprise yourself when you fall in love with someone completely different!