Do we (the humanity) have any satellites in the Sun-Earth $L_3$ point? If not, then what are the plans to put some ships into this point?
TL;DR No, there are no sats there today, and no declared plans from any of space agencies to do that.
An Earth-Sun $L_3$ point is an unfortunate place for a satellite to be in.
First off, it is unstable (thus without constant station-keeping burns an object placed there will fall out and start roaming about the Solar System). An alternative would be a halo or Lissajous orbit around $L_3$.
Second, the satellite would face significant difficulties in communicating via DSN. It would be permanently in solar conjunction with the Earth, and as the Sun is a very strong source of radio noise, the signal would drown in the noise. Of course, one can use satellites placed in Earth-Sun $L_4$ or $L_5$ points as relays, but it would make the whole $L_3$ program much more costly and unreliable.
This leads us to the third point: what can be the mission requiring a halo/Lissajous orbit around $L_3$?
Space telescope? Nah, there's nothing in the far space that can't be observed in the course of Earth's normal yearly travel around the Sun.
Solar weather observations? Nope, by definition we are interested in solar events going our way, not in the opposite direction.
There's only one speculative mission that forces us to have something in $L_3$: watching out for an alien invasion, 'cause an advanced spacefaring civilization would definitely want to reduce our reaction time and choose an approach obscured by the Sun. Even then, $L_4$ and $L_5$ points offer better link budgets and only slightly worse detection conditions. A smart alien invasion would seek to disrupt the early warning network by inducing a fault that could be attributed to natural causes.
To sum it up: an $L_3$ satellite is not worth it unless you believe in evil aliens wanting to invade the Earth.
EDIT: Following andy256's suggestion, I quote the para from Tantardini et al. (2010) paper:
A space observatory placed at or around L3 could provide insight into local astrophysical phenomena such as the solar wind and the properties of the Sun’s magnetosphere; it could monitor the evolution of the sunspots, perform solar storm forecast and observe the space environment from a new perspective, hidden to the Earth, and in this way supplement and complete the information obtainable, e.g., from L1 or from Earth based observatories. A satellite at L3 could be even thought of as part of a circular or spherical constellation of s/c monitoring the Sun’s activity and the space environment at many different angles. Moreover, L3 may constitute a privileged site to perform relativity experiments, such as measuring the gravitational bending of light on behalf of the Sun, as a follow-up of the Cassini–Huygens radio science observations. Finally, some minor bodies such as NEOs and comets, hidden by the Sun as viewed from Earth, could be observed and even tracked from this more favorable location.
Please note that I'm not convinced in the slightest by their arguments.
Currently NASA has two satellites flying in Solar conjunction with Earth, near the L3 point of Sun-Earth - STEREO A and STEREO B. They will not stop at L3, and the distance to L3 may be rather great. Current position ("Where is STEREO Today?"): http://stereo.gsfc.nasa.gov/where.shtml
Some information about mission: https://en.wikipedia.org/wiki/STEREO
STEREO is the mission to observe Sun from two additional angles. One satellite was started in 2006 to orbit, lower than Earth, and other - to higher orbit. One is "ahead" (A) of Earth and other is "behind" (B) of Earth with the angle Satellite - Sun - Earth increasing at rate 21.650 degrees per year for A and decreasing at rate -22 degrees/year for B.
they passed through Earth's Lagrangian points L4 and L5, in late 2009
In 2015 both STEREO satellites are behind Sun - http://www.nasa.gov/content/goddard/stereo-entering-new-stage-of-operations/:
- "A" - from March 24 to July 7, 2015
- "B" - from Jan. 22 to March 23, 2015
In 2009 they where two artificial satellites made closest pass near L4 and L5 - http://www.nasa.gov/mission_pages/stereo/news/gravity_parking.html
It will take several months for STEREO to pass through them, with STEREO A making its closest pass to L4 in September, and STEREO B making its closest pass to L5 in October.
In time, all three (ES L3, L4, and L5) are likely to become viable locations for stationary, halo, or lissajous orbital satellites, largely in part to our current desires to place humans on mars and to mine the asteroid belt. As we all know, Earth and Mars are not synchronized space swimmers, so having distant communication devices that would enable continued contact for the 1-way adventurers to mars would prove evermore compelling. Eventually (100s of years from now), space stations placed at L3, L4, and L5 should also be of interest when considering where to accumulate mined asteroid ores, with possibilities of refineries in those locations as well. All this with the distant future in mind, of course (and nothing any of us will ever see, unless cryogenic freezing becomes a thing hahahahaha).