Let's forget about what's possible physically, it doesn't matter for this discussion.
The "issue" is that even mathematically there is no rest frame for a lightspeed particle. In relativity, the main point is that it doesn't matter who's moving: If I'm moving for an observer on earth with velocity $v$, then for me, the earth is moving with velocity $v$ (talking about magnitude, not direction). But if you try to "sit on a photon" and look around, you get into trouble (the whole universe would need to travel with lightspeed from perspective of a photon). Still, you can make sense of it if you take the limit carefully.
Here it goes:
A lightspeed observer perceives no time passing from beginning of their journey to the end. The universe is also infinitely contracted, it's a 2D pancake, there is no distance to travel, everything is here. That means, that the beginning and end of your journey are the same point in space and the same time. It's basically teleportation for you.
When a photon is emitted from your light source, it "travels" to the screen when it is absorbed (or reflected, re-emitted, however you want to see). But the photon doesn't live through that. He's born and dies at the same time, it didn't age or feel time, and with regard to it, the light source, the screen, and every point in between, are one and the same point, where he exists for that single moment in time.
If you turn this around to the external observer: you can never see a particle traveling at light speed, without intercepting it and effectively stopping its journey, because it has no time to emit anything, so it can't do anything on its own without hitting another particle (scattering). You've never seen light travelling at lightspeed, you can only see its arrival.
Taking the limit again: if you consider a very fast, but not quite lightspeed particle, whatever it emits, will be relativistically beamed in the forward direction (angle aberration and red/blueshift), so that it's redshifted to almost nothing in backward direction and blueshifted + aberrated in the forward direction so that it arrives barely before the particle does. In the limit, it would all arrive at the same time in the same direction, effectively meaning that it all just travels together (so nothing happened, there's no difference if it splits apart or if it doesn't, it's all just one chunk).
Mathematically speaking, you have to consider a space-time distance between two events in Minkowski metric, which is $s^2=(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2-(ct_1-ct_2)^2$. This is zero if you stay on the light cone (points in space-time that are connected with a light ray), which is a theoretical restatement of "for photon, the beginning and the end are the same".
To think about it another, stranger way: light circumvents the concept of distance and takes shortcuts through time and space, connecting distant events as if there was no distance at all, but can only do so for events, that for an external observer obey the distance=lightspeed×time connection. This makes the space-time strangely connected with specific rules: when you send light signals to some other place, the information from perspective of a photon instantaneously teleports in space and time to the final destination, but because it also teleported in time (relative to our external observer coordinates, which are not squished into a pancake), we must still wait. In curved time-space, for example, because of massive objects, these "shortcuts" are not straight, but they are still "instantaneous" for light - it's just criss-crossed in some curved manner so you can't intuitively tell which space-time point light will "teleport" to by simply drawing a straight line. Notice that in curved space-time light can get stretched (redshifted) during flight. That seems paradoxical at first, because nothing can happen to light when it's travelling, because it feels no time. However, there is no paradox, the light is the same, it still just went "poof" from source to the final observer, the catch is, that the observer is "playing it back" at a different rate than the source "played it", because their times don't run at the same rate.
The infinite mass thing is just a "helper" from non-relativistic mechanics, it actually isn't a sensible thing to say. Mostly, mass is what we call "rest mass", and thus is constant. Energy $E=\sqrt{(mc^2)^2+p^2}$ is the one that depends on velocity ($p$ is momentum), and if your mass is nonzero, you would need infinite energy to reach speed of light. Note that the mass $m$ doesn't change, only the momentum does, when you accelerate.
