I want to replace my forum name and the smf logo with my own header/banner and I hope I'm right in saying that the easiest way to do this is via the current theme settings and inputting a logo image. I've had a play around with it and altered the size several times and it never comes up just right - so is there a standard size for this image using SMF 1.1.13 and the Default Theme - Core?
No standard size. Reason being is that not everyone is using the same resolution. You make it work and look pretty on your browser at 1024x768, then someone comes along on a much larger or smaller resolution and it looks borked for them. If I were you I'd just pick a size that looks best for you. Unless you want to make some type of logo that is able to repeat so that no matter the resolution it will repeat and fill the voids.
Or use a small image so that it fits everywhere
Well it depends which computer I'm on as to what resolution I'm working with - currently on 1920x1080. But presumably it optimizes for each user of my forum so it doesn't matter what resolution they have? I want the banner to go the length of the top of the page, not just be a small image.
Then you have to redesign the them header to be able to repeat the banner or something. You can't just slap one image in the header, and have the image expand or shrink to fit the page according to resolution. Well, not without degradation of the image anyway.
So does that mean that logo image box in the current theme settings area is not really the way to go about it - that it really needs a bit of code to put in?
Yes. The logo option only allows you to put a single image in there. You'll have to make some code edits to get it to repeat the image, or to put additional images to repeat on the sides to fill the gaps or something. There's a lot of different ways.
How would I do a banner something like this:
http://simpleportal.net/index.php
Not something that can really be explained in a post. If I were you, I would go through themes on the site and find one that does what you need. Then take a look at the code to see how it's done. If get stuck on something or have a question feel free to post and ask about the specifics of it.