I do not have a table saw and for some unknown reason I am scared of them.
Some fear is entirely appropriate, table saws are potentially very dangerous tools. The table saw regularly tops lists of the most dangerous power tool and it may account for the largest number of serious injuries in a woodworking shop.
Anyway, you don't need a table saw for doing picture frames rest assured! Picture frames were made for a long long time before table saws came on the scene.
Even today if someone owned one they might not use their table saw for doing picture frames, except for some initial ripping cuts, because they wouldn't be the tool of choice for forming the rabbet/rebate or cutting the mitres.
I am planning to use a hand saw (with miter box) and router. I was hoping to be able to find 1” x 1” square dowel. Remove enough material with the router to be able to install the plexiglass.
This is entirely doable. Be aware though that sawn mitres might not be quite perfect and for a very good fit it would be common to trim the mitres, e.g. usinig a hand plane and a mitre shooting board, see here and here.
There is another option to consider which would make the construction even simpler, and that is to build up as covered in this previous Answer.
Where can I find 1” x 1” square dowel (it would be nice if it was walnut).
Can't answer that for you I'm afraid as it's completely dependent on location. But in many places walnut in pre-formed shapes won't be available. The usual route would be to buy wider boards and then rip narrow strips from them which are then formed into the frame members.
Other woods, including poplar, pine and white or red oak are commonly available as readymade strip material however. It isn't uncommon to get a lesser wood (e.g. poplar) and then colouring it so it looks like walnut, either with wood stain and then applying a gloss finish, or by using "gel stain" (a type of coloured varnish).
The frame in the picture below is walnut. What kind of finish did they apply?
Impossible to tell, it could be nearly anything. (Incidentally the frame in the photo actually looks like it might be oak.)
But if you're after an easy, reliable way to finish that gives a good shine then I highly recommend using wipe-on varnish. They sell "Wiping Varnish" already made up but there's no reason to buy it and pay too much for it, you can easily make it yourself at home. Buy regular gloss polyurethane varnish, decant a little into a jar with a tight lid, thin it with some additional mineral spirits (1/3 works fine but you can make it as thin as you like). Shake the jar and it's ready to use.
You can wipe the diluted varnish on as the name suggests or brush it on, it doesn't matter. You don't have to but some or all of the excess can be wiped from the wood and then it's left to dry. It dries faster the more you wipe away but builds up more slowly.
Safety note: if you wipe on with cloths or paper towels these should be laid out flat to dry and not left scrunched up.
Three coats will usually prove just enough to finish hardwood so it looks uniformly coated, but by all means apply more coats if you want or the wood looks like it needs it.