Trumpet vine! Yeah., that's a very invasive plant. Took us several years, (maybe 5), to actually get rid of the one plant we had put in. Sucker kept on coming back.
The wall problem you have seems pretty serious and with a shoestring budget it's even tougher to solve.
A shovel and a strong back is probably going to be the best solution. Any chance of renting a small excavator for a weekend? Maybe hire a few hardy under the table day laborers? Most towns have a place you can drive by and pick up a few workers on any given morning.
Then just replace the whole thing.
If that isn't in the budget you could try this.
The only thing I can think of is to get some steel galvanized pipe like the black gas pipe. You can usually purchase 4' precut sections with threads on each end for end caps. Get some 5x5 or 6x6 steel plates with holes in them to accommodate the pipe. You might have to get a metal shop to cut and drill the plates for you. Get a post hole digger and drill a series of post holes about 3-4 ft behind the wall, about 2 - 3 ft apart. Drill a hole in the front of the ties, even with the post holes. Take a pipe and insert into hole, slip on plate, screw on end cap. Now sledge hammer the pipes through the hole in the ties and into the hillside until the plate is flush against the tie. You want to measure the holes so that the pipe ends on the backside of the posthole. Put on the end caps once they are in the post hole. You should maybe have 2-3 pipes come through each post hole. Now fill the holes up with concrete. On the tie side you would have a plate and the end cap on it. In the post hole, a solid post of concrete that the pipe passes into with the end cap that prevents it from sliding out of the concrete post.
This idea isn't much different then they use on old row homes here in Philly. When a middle home collapses or burns, and the house is torn down, they often run rods through the joist spaces and to the outside of the house where large plates and bolt are bolted on. This helps keep the walls from falling down since the wall wasn't originally an exterior wall.
Also see this link for suggestions:
http://www.ehow.com/how_7536569_retain-walls-made-railroad-ties.html
You can also maybe reinforce the front wall by drilling a vertical hole down through the ties and pounding a some long rebar down through the ties and a foot or so into the ground. Do this every 2ft or so along the wall.
I don't think there is a good solution though for the rotted piece of wood. If its only 6" you might be able to reinforce the wall first, into hill, then cut out the rotted piece. Use a small bottle jack in the space to lift the tie(s) above it, (cut a piece of oak to place between the jack and tie so the jack doesn't just punch through the tie), lift the tie just a tad, enough to put in a small metal shim. Custom cut a new piece and slip it in. Remove the shims. drill a hole through the ties above and through the new piece. Drop in some rebar. Cut and seal the hole with some epoxy or plastic wood.
This might not be a perfect solution, but it will probably last as long as the ties will since it sounds like they already have quite a few years on them.
Craig