Welcome to The Helpful Gardener. Sammyd has answered your questions, and with no disrespect to him, I don't totally agree with all that he recommends.
Overall size of the garden I plan to build will be 6' x 8' x 18". ... using 2x4s and a fine meshed wire screening. Are my dimensions adequate to grow a decent variety of vegetables & herbs?
Not sure what you mean when you say "adequate", but you can have variety depending on what you grow and how carefully you plant and harvest and plant again. I do agree with Sammyd about using a larger deminsion lumber. Since you want to make it 18" deep, I would suggest you use 2X10s. You can probably get them in 8' lengths. If you will be able to reach from both sides of the bed, 6' wide would be ok as you will be reaching into the beds up to 3' on each side if your arms are long enough without stepping into it. Generally a bed 4' wide is best though. That way you only have to reach in 2' from either side and it's more doable for most folks. You would be able to cut those 2x10x8s in half for your depth.
Another option would be 2x12 lumber. Click on the picture for a better view.
What is the meshed wire screening for?
What kind of wood should I use? Treated/Untreated?
Untreated! Red cedar and redwood are the longest lasting woods you can use that are untreated. Here's some info on using treated wood in the garden which is an unhealthy idea.
Also, what kind of fertilization do you reccommend?
Organic. Use a ratio of 60% screened topsoil to 40% compost. Your veggies and herbs will love you!
The planter box will start on my patio which gets half shade & half
sun during the summer, and extend 8 feet over the lawn, being raised about 6 inches as to not kill the grass.
I can't imagine the grass surviving if the planter will shade the grass, even if it is elevated. You'll end up shading out the grass if it extends 8' over the lawn. Why not just make the planter deeper there for the roots to go into the native soil? You can break up the soil in that area and add 3" or 4" of compost to improve drainage and texture.
You can build a trellis like one of these for your peas, beans, cucumbers, etc.
You will find that your tomato plants will be about 3' wide at maturity and will need most of the spring and summer to mature and bear. Garlic is generally planted in fall and harvested the following summer. If you start your onions from sets they should mature in time for harvest. Here's some veggie growing guides you should find helpful.
I like Sammyd's suggestion of growing the herbs in pots. Mints should definately be grown in pots as they will take over the garden in no time. You can sink the pot into the soil for the winter if you like since it's perennial. Chives are also perennial so a corner for them might be ok, just don't let them go to seed or they too will be everywhere. Sage might be perennial for you but it tends to sprawl so you'll need a bit of room for it. You might find this helpful for growing herbs.
You should find these sites helpful.
Since your garden will be small, this first site shows good and bad companions. The other lists good and bad veggie and herb compos.
Also consider square foot gardening.