Could we place a satellite in the wake of the Earth's orbit so that when the Earth completes its orbit of the sun it would effectively meet back up with said satellite?
No. At least not the way you're thinking of.
The reason that the Earth orbits around the Sun is that the Sun's gravity is always pulling it inwards. This is not unlike an object being swung around in a circle on a rope: if you do this, you always have to pull the object inwards to maintain the circular path.
A satellite that was somehow made "stationary" in the solar system would still feel the Sun's gravity, but wouldn't be moving around in a circle. So it would just fall straight into the Sun. You could, if you wanted to, put a satellite on an elliptical orbit that had a period of one year. This satellite would leave Earth, travel on its own orbit through the solar system, and then would happen to meet up with Earth a year later. But this wouldn't be a "stationary" satellite.
If you place a space probe in the wake of earth, outside earth sphere of influence, it will just follow the earth, orbiting at a similar speed, never reaching the sun.
You cannot orbit the earth and remain in it's wake (you wouldn't be orbiting).
To reach the sun from earth, you would need to cancel the speed of your orbit around the sun, not "orbiting in it's wake"
Yes, but it wouldn't be a traditional satellite.
The Earth is constantly being pulled towards the Sun by the Sun's gravity. It orbits because it's also moving sideways fast enough that gravity only acts to bend its motion into a circle. If you launched a satellite in a way to cancel the speed it had from being on Earth, it would fall into the Sun in short order.
Since your probe isn't able to stay out of the Sun by moving sideways, you need some other way to deal with the Sun's gravity. Fortunately, there is such a way: the solar wind is a constant flow of high-energy particles moving away from the Sun, while light itself can exert a force on objects. You can harvest these forces with a magnetic sail or solar sail to let your probe stay in place while waiting for the Earth to meet up with it again.
Short answer: Yes and No.
Long answer: You can't simply "place" a satellite outside of Earth's SOI (Sphere of Influence). If you do, it'll continue moving along with earth. However let's just say we have a powerful rocket that can slow us down by 30 km/s (Earth's orbital speed) so that the satellite remains stationary. The satellite won't stay there forever, instead it'll slowly fall to the Sun. However, what you can do is put a satellite into an elliptical orbit around the sun that takes exactly 365 days to complete on orbit. That means, once the satellite completes one elliptical orbit, the Earth will also complete one orbit meaning they'll meet up again.