In the Double Slit Experiment, what is the longest time $T$ recorded between shots of particles, electrons, photons, etc.? How can we be certain that electrons fired do not leave some kind of residual interference for the next shot?
What is the longest time recorded between shots?
 A: We know that there is no residual interference because the pattern does not depend on the size of the interval between photons. If you load up the slits with a really bright laser you get the same thing if you gradually turn down the intensity.
Also, the pattern does not depend on the material of the slits, so long as the required opaque parts are opaque and the required transparent parts are transparent. All that matters is the geometry. So any possible residual does not hang around on any particular material. Iron, carbon, the edge of your finger, fibers of hair, photographic emulsion from old-style photographic film, medical hypodermic needles, all of these produce the expected diffraction pattern.
Also, shining one laser in this direction, and another in that direction, makes no difference. So any residual interference would have to have a memory for directions.
Also, shining a red laser and a green laser at the same time makes no difference. So any residual inteference would have to have a memory for wavelength.
Also, diffraction works as expected for widely different wavelengths. Everything from quite long radio waves to quite hard x-rays diffract as expected. So any residual interference would have to function over this range of wavelengths.
So this increasingly unlikely residual would have to work for arbitrary intensities, last for very long times, spread over the entire length of the slits, only work for photons of the exact right energy coming from the correct direction, work for slits made of any material, and work for energy from many-meters radio waves to many-MeV x-rays.
Veritasium did the experiment with about 100 photons per second. This gives an average distance between photons of 3000 km. That is, since his equipment was about 2 meters long, there was less than one chance in 1 million of any given photon overlapping with its neighbor. He had a lot of background (about 7 photons per second) polluting the pattern but he still got it. He has better equipment than my highschool did, but it wasn't all that good.
A: Two-slit experiment is mainly a pedagogical device to teach the basics of QM, demonstrating the wave nature of electrons. In this sense it is better studied alongside the two-slit experiment for classical light (aka *diffraction on a slit). Many of the restricting assumptions, such as those concerning the particles being non-interacting or obeying Fermi-Dirac /Bose-Einstein statistics can be lifted, and indeed studied - e g., in solid-state Aharonov-Bohm interferometers. In fact, there is a range of peculiar phenomena appearing in such interferometers, having to do with their realistic realization - phase rigidity, phase lapses, etc. - some having simple explanation and others having required deeper research to understand.
A: In the typical double slit experiment the electron source is a cathode , even for single electrons at a time, and thus an incoherent beam, so your "residual interference" cannot be from the incoming beam. In your comment you explain:

I just mean how can we be certain that a particle fired doesn't have an impact on its environment, which in turn could impact the next particle.

The size of the electron footprints , where all the energy is expended on the atoms of the screen in dimension of microns, a good distance apart preclude the residual interference to come from the screen. Any residual interference will come from the "slits" . If you read the link you will see the experimental set up for the single electron at a time experiment.
For the single photon double slit I agree with the answer of Dan, and would add that if the single photon had  left an appreciable imprint on the slits,it would lose energy and change frequency, which is not observed. I suppose a limit could be calculated by the fact that the frequency is invariant.
A: You assume that spacetime is plastic, like skiers carving paths through the snow. I was told that this is not the case, in terms of current theory.
On the other hand, assume that the insides of the double-slit device are evacuated. Shoot a photon, let it collide with the sensor array. Then, flush the device with gas (like air), and re-evacuate before shooting the next photon. Does this affect the pattern on the sensor array over time? It was worth thinking about anyway.
