Gravitons are affected by gravitational fields, and viceversa.
From the classical Einstein Field Equations point of view it is straightforward. Those equations show even as an approximation that there are terms of the metric which go as 1/r at large distances, and those are the strongest surviving terms we would call gravitational radiation. Or gravitons. We can treat them linearly to determine their value (the coefficients of the 1/r terms), and it turns out that is what we are now able to detect in LIGO as they affect the distance between mirrors in the interferometers. So, gravitons indeed affect spacetime, i.e., cause gravity.
The other side is they they also are affected by it, they have to travel in the spacetime as null geodesics, i.e. Like light, to first order (I mean, they will affect that spacetime also, but that is a higher order approximation you can at first ignore). For LIGO we had to take the cosmology metric and from it calculate what redshift it would have with respect to us - Ie, the shift in freq from radiated to what we received.
Yes, it all has to be self consistent, but you can calculate often approximately right to the levels of accuracy we are able to achieve now.
As for gravitons interacting with other gravitons you'd need to do that extra higher order calculation - calculate how much the metric changes spacetime, and then the change of geodesic orbits due to that. We've not gotten to that level of accuracy. But in calculating how much gravitational wave gets produced we do go to various orders of approximation beyond the Nwtronian level, it has been parametrized to levels called eg PPN2 or PPN2.5 or higher. One of the effects we calculate is that as the grav waves get emmitted by the merging black holes it causes a recoil on the resulting leftover black hole, could be 5000 km/sec or more, just because of the conservation of momentum from emitting the gravitational energy preponderantly asymmetrically. Enough momentum to get merged black holes kicked out of their galaxies in some cases.
So, yes, all kinds of effects, and it depends on the level,of approximation you need to account for.
See the Wikipedia article, it mentions the recoil towards the end, at
There's others, Google recoiling black hole mergers.
Finer or more subtle interactions we are still not able to detect