Can photon be said to warp time only and not space? Can SR be modelled like GR, considering warping of temporal dimension only? If we try to get closer and closer to a photon (i.e. get closer to speed of light), time slows down in a fashion similar to when we get closer and closer to a massive object. Are these two things related somehow?
Following up on this, can we say the following:

*

*Normal matter -> bends both space and time

*Photons -> bends time only

*Dark matter -> bends space only

It is said that dark matter does not interact with electromagnetism, but has mass. Can this be because dark matter bends space only, while photons bend time only and hence they somehow can't interact?
 A: Nope photons doesn't just bend time only. It bends, very bad word, space also for example radiation dominated universe (FRW metric). Why? Cause photon has an associated $T_{\mu\nu}$ which tells us how $g_{\mu\nu}$ changes, or in sloppy language how's space time bends. Same goes for dark matter.
A: Let us take the different concepts appearing here apart and see if it answers your question. First of all general relativity (GR) is a generalization of special relativity (SR), the main difference being that in GR the space-time (notice they are pasted together, they are one thing) is curved by energy (Check Einstein field equations). Both SR and GR are built over the idea that frames of reference are connected by Lorentz transformations instead of Galilean transformations, in other words the speed of light is implemented as the speed limit of the Universe (for anything).
Under this view we can now group the things you mention. Matter, photons and dark matter all have energy, they are just different presentations if you will, but to all of them we can associate an energy density, therefore they all curve (space-time). The question is how much. How much they curve space-time depends on their masses (or energy densities more formally). So if you are speaking about a single photon, you can very likely neglect the curvature in your problem (exceptions may occur depending on the relevant scales). Same goes for matter, you need large amounts of it though to curve space-time, like the amount in a Start or a Black-hole to witness any affect of curvature.
Dark matter is a more subtle topic, but one thing we know and it is that it has mass and interacts gravitationally just like matter does. So in terms of curvature it has the same effects. Historically it is exactly this behaviour that lead the community to coin the name Dark matter since, we can observe its gravitational effects, so it must be some sort of matter, but just can't see with light, thus "Dark".
So you cannot get "closer and closer" to a photon, the photon is playing no role. You can't just travel at the speed of light (while having a rest mass), without investing infinite energy accelerating to that speed.
All in all, the message really is space and time are "glued" together, nothing "bends" just one of them. I suggest you try to look at the mathematical details if possible to understand what a metric (in GR) means and then study what space-time intervals are.
A: The premiss of the question is flawed. Time does not slow down as you approach the speed of light because you cannot approach the speed of light. No matter how you move inertially, spacetime appears the same to you, in particular: light moves at $c$ in all directions, which means: you cannot approach the speed of light.
With 2 frames of reference, velocity makes sense, but each observer sees the other's clock tick slower, which is described in the usual manner with titled hyper-planes of simultaneity.
