If the metric gμν is dimensionless and gravitons are quantum excitations of the metric does that mean that gravitons themselves are dimensionless?
It has dimensions , at least look here to a particular metric , the matrix elements have the dimensions of meter square.
Is graviton energy included in the stress-energy tensor Tμν?
In phsyics there are several frames where different theories describe mathematically the data and observations, and also predict new systems, but in mainstream physics they are consistent with one another in the region of overlap.
Metrics of General Relativity belong to large dimensions large masses . GR is a classical theory, which means it is not quantized. The graviton is the hypothetical gauge boson of a quantized gravitational theory, expected in the future. At the moment only effective quantization of gravitation is used in mainstream cosmological models.
So it is mixing up apples and oranges to require GR to have gravitons included.
Actually classical gravitational waves can be detected so does that imply that gravitons can't be dimensionless?
It is not clear what you mean by dimensionless. All the particles in the standard model are point particles , thus have no dimensions. They are described when interacting with an energy and momentum four vector.
In string theories, which can quantize gravity, the graviton is part of the particle spectrum. If a definite theory is found then the graviton will be like the photon a zero dimension elementary particle.
Note that it is expected that mathematically, the quantized gravity theory will reduce to GR for large masses and dimensions. Here one can see the equivalent for electromagnetism, how classical electromagnetic fields emerge from the quantized fields of QED.
This emergent behavior happens in regions of phase space overlap between theories, for example: thermodynamics emerges from classical statistical mechanics. Otherwise physics would not be a consistent theoretical system.