Best way to fill gaps between flooring panels evenly
I'd strongly recommend talking to the landlord to see if he will fix the improperly installed laminate flooring.
If he refuses, but gives you permission (I'd get it in writing, just to be sure) to make a permanent change to the apartment, I'd recommend a two step process:

Squeeze some caulk (clear or matching color if you can find it) into the gaps.

Silicone will probably last longer than an acrylic/latex and is more likely to be available in a clear &quot;color&quot;.
You'll need to barely cut open the tip of the tube - you'll want a very fine bead so you can get it to go into the gaps, not just sitting on top.
You may need to use your finger or a plastic tool (a plastic spoon would work well) to push the caulk into the gap.
Check the tube for the specific curing time for the caulk you purchase. It may say that it's paintable in 20 minutes - that's not what you're looking for. You don't want the minimum set time, you want it to be pretty well done. It will probably be 24-hours or longer.
As you pipe the caulk in, you will get a ridge above the surface, that's where step 2 comes in.


Using a sharp razor blade, very carefully cut the excess caulk off the surface.

Hold the blade 90° to the gap you've filled and nearly flat to the floor. I'd suggest holding the blade as flat to the floor as you possibly can - the goal is to cut off the excess caulk, not cut the floor.
You want to cut along the length of the gap you've show (&quot;horizontally&quot; as you've described it), not along the width of the gap. The edges of the blade should be on either side of the gap, parallel to the nice tight joints that you're not fixing.
Push it along the gap to cut the bead that's sticking up.
Note that if you move the blade side-to-side in a cutting motion, you could cut into the surface of the laminate causing more damage than you're fixing.
Note that if you hold the blade too steeply to the floor, you could cut into the surface of the laminate causing more damage than you're fixing.
You might pull up some caulk. If this happens your blade probably isn't sharp anymore, go grab a new one (blades are cheap compared to the labor of re-fixing, don't hesitate to replace more frequently than less frequently), then redo the caulk for this portion. I'd save all the &quot;re-dos&quot; for the end, you may have more than one spot to repair.



Note that if you use silicone, it will leave a slightly greasy feeling anywhere it touches. Any amount of silicone that gets into the texture of the floor instead of just into the gap will be difficult to get out of the texture. It will wear off over time, but I'm not sure exactly how to define &quot;time&quot; - it could be a matter of a couple of days for high traffic area to months or more. The caulk is, after all, designed to stick to a variety of surfaces and not come off - that's how it does its job of sealing. It's possible probable that after it's fully cured (a few days) you'll be able to rub this thin coat off using a finger or pencil eraser.
I'm not quite sure how slippery it will actually be underfoot, but if you're sensitive enough to feel these gaps in the flooring (I can definitely understand that), you're probably sensitive enough to feel the greasy feeling and may not like the result.
Note that both silicone and acrylic/latex caulks will have an odor to them. They're not horrible nor toxic (check the tube, just to be sure), but you will want some ventilation. Open windows and turn on fans blowing air out of the apartment while you're working and as long after as you can. I just did some fresh silicone caulking around my tub and it took about 14 hours for the odor to disapate enough that I couldn't detect it - other family members couldn't smell it earlier than that. If you have a sensitive nose, you might smell it for up to a week. Acrylic/latex caulks will be similar.