What would the wavelength of the Cosmic Background Gravitational Wave radiation be? Considering electromagnetic CMB can only see light as old as 380,000 years after the Big Bang, whilst theoretically those being gravitational should be formed from the beginning, what would their wavelength be, and do we have the technology to detect them in the foreseeable future?
 A: There is much less understood about what kind of "cosmic background" may exist for gravitational waves than for electromagnetic waves. The CMB comes from a specific epoch -- when the universe became transparent to light, and the electromagnetic field, previously in equilibrium with matter, was "released" as a self-contained thermal spectrum at ~3000 K (which has redshifted to ~3 K at the present day).
The universe has been transparent to gravitational waves since much earlier. There is unlikely to be a thermal gravitational background, because gravitational waves never had time to equilibrate in the first place, due to the weakness of gravity. Notionally, if a thermal spectrum had redshifted since the Planck epoch, it would be of the same order as the CMB today, ~1 K in temperature and ~3 mm in wavelength. Contrary to joseph h's answer, these hypothetical thermal gravitational waves would have too short a wavelength (too high a frequency) to detect with known technology. But as noted, they (thermal gravitational waves and not traditional gravitational waves) are unlikely to exist anyway.
The actual gravitational background likely does not have a simple cosmic origin like the CMB, but consists of a superposition of waves emitted from many sources, at many frequencies, over the life of the universe. So there is no simple formula for its spectrum.
It is relevant here that the angular resolution of current gravitational wave detectors is very poor. Thus, a random-seeming background could easily be formed from a combination of many sources. It is like a very nearsighted person looking at the night sky, and seeing only a faint general glow from the out-of-focus stars. As gravitational wave detectors get better resolution, what was observed as background will increasingly be observed as specific sources.
A: *

*What typical wavelength would a gravitational wave background have?

The answer is: all frequencies up to some cut-off scale, and above that not much. So there are long wavelengths and short, but not very short. The details are not known but some models suggest a scale-invariant spectrum, which means you have equal amounts of power in each fractional increase in the frequency range, up to some finite range. One does not expect extremely high frequencies, but they could go up to frequencies such as GHz.


*Can such cosmic background waves be detected?

Direct detection not feasible into the foreseeable future, but a gravitational wave signature might be detected in the pattern of polarization of the CMB, and the detection of this is just on the margin of what has already been done. That is, it has not been seen yet, but might possibly be seen soon. The idea is that one asks what kind of physics could give rise to certain kinds of polarization pattern, and the answer for some quadrupole-shaped patterns is that gravitational waves in the early-universe plasma are the most likely cause. One expects gravitational waves in any sufficiently turbulent matter; the trouble with detecting their effects in this case is that the early universe was remarkably smooth so the effects are especially small.
For more info you could try a paper I wrote which is related to this: https://arxiv.org/abs/1710.05816
See also:
Is there a Cosmic Gravitational Background Radiation (CGBR)?
A: It is unlikely that we can detect gravitational waves from the Big Bang with current technology. Due to universal expansion, such waves would have very large wavelengths.
We would need interferometers that are thousands, perhaps millions, of kilometers long to detect them. The LIGO observatory simply would not be sufficient. To put things into perspective, consider that the arms of the LIGO interferometer have a length of $4km$. But it is important to note that the effective LIGO arm length is $1600km$ (the light beam inside the interferometer is reflected back and forth 400 times) and LIGO is most "sensitive" at a frequency of about $150Hz$, which would correspond to a wavelength of $\lambda \approx 2000km$ meaning the LIGO arms have a length of $\approx\frac{\lambda}{2}$.

do we have the technology to detect them in the foreseeable future?

There is a proposal called "LISA" that  will involve a system of satellites in space separated by large distances ($\gt 10^6$ km), that could possibly detect gravitational waves (gravitational waves that where emitted before the photon epoch - up to 380,000 years after the Big Bang as you mentioned).
From that link:
"The Laser Interferometer Space Antenna (LISA) is a proposed space probe to detect and accurately measure gravitational waves—tiny ripples in the fabric of spacetime—from astronomical sources. LISA would be the first dedicated space-based gravitational wave detector. It aims to measure gravitational waves directly by using laser interferometry. The LISA concept has a constellation of three spacecraft arranged in an equilateral triangle with sides 2.5 million kilometres long, flying along an Earth-like heliocentric orbit. The distance between the satellites is precisely monitored to detect a passing gravitational wave...
Potential sources for signals [that LISA could detect] are merging massive black holes at the center of galaxies, massive black holes orbited by small compact objects, known as extreme mass ratio inspirals, binaries of compact stars in our Galaxy, and possibly other sources of cosmological origin, such as the very early phase of the Big Bang
, and speculative astrophysical objects like cosmic strings and domain boundaries."
