I put D'addario EXL110 strings in my 25.5 scale guitar (Their gauge is .10 - .46).

When I tuned my guitar in E standard, I noticed in my guitar tuner that when I pluck the sixth string, the pitch goes a little sharp and then returns to E after some milliseconds. If I lower the tuning a little, then, when I pluck the string, the pitch is E, but after some milliseconds, the E becomes a little flat. This effect I'm describing is more pronounced if I pluck the string strongly.
I noticed this effect is greatly reduced in the first three strings (G, B, E).

Does this mean that if I pluck the strings hard, I will play out of tuning? Should I play all the strings with the same amount of strength, so that all the notes are in tune? Also, does this mean I should put a thicker low E string in my guitar?

.46-.10 is perfectly fine for a 25.5 guitar in E standard. you can go up to .48 to see how it feels but this is going to happen no matter what. for regular playing, generally, just tuning it so that it evens out at E is fine. when recording, a lot of engineers suggest tuning to the riff, ie: tuning so that the initial attack is in tune for riffs with a lot of playing, tuning so that the sustain is in tune for riffs with a lot of sustained notes.

If you use large picks and/or let a good amount of them stick out, you might also be forcing the string to bend around and snap back from several mm of pick. Maybe try using a jazz pick and let ~2mm of it stick out.

For what it's worth, there's no reason to be opinionated or rigid when it comes to this stuff. It's fine to experiment and use whatever seems to work for you. If you changed string gauge and now it's hard to intonate or doesn't stay in tune, or feels weird or difficult to play, it was probably not the right move. There's no right and wrong except what works for you. Antti Boman claimed to have recorded Demilich's Nespithe in A-standard with 9-46 strings on a Gibson Explorer. SRV played Stratocasters in Eb-standard with 13-58 strings.