When pronouncing an 'l' (alveolar lateral), my tongue stays on alveolar ridge of my mouth (the roof, right behind the teeth) for the entire time. When the 'l' sound ends, my tongue leaves the roof of my mouth, but it never "strikes the bottom" of my mouth. If there's a vowel next, it moves to a neutral position for a vowel. If the 'l' was the end of the speech, it just kind of sits in place for a second after the noise stops. If it's followed by another consonant, it moves for that articulation (e.g. when saying "all the sounds", it moves directly from the alveolar ridge to the teeth for the dental fricative).
When pronouncing an 'r' (alveolar approximant), I, and I think most English speakers, roll the tongue backwards to some degree (although not enough that the point of articulation is the bottom of the tongue). I.e. the tongue does not "stay at the bottom". Yes, it is certainly possible to pronounce an English 'r' with the tip of your tongue held down (unlike 'l'), but I think most English speakers would find this "impossible" at first, and then with a minute of practice would realize that it was possible but difficult. In English, these two articulations are perceived as the same phoneme; I assume that some language somewhere differentiates between them. I notice that often when I speak (English) to natives of India (whose first languages I have not identified), I get the sense that they have a richer complex of 'r' noises than I do.
In summary, "l" does not involve "striking the bottom [of the mouth]", and "r" typically does not involve the tongue staying at the bottom of the mouth, although it can.
Is that more clear? I agree that there are other r and l sounds in English, but there aren't in Japanese, there is a fixed set of phonemes. I'm trying to explain how each Japanese l/r sound is clearly split into a different l and r sound in English. Of course, there are many English l and r sounds that are badly approximated by the five l/r sounds in Japanese.
Anyway, my point was this: your description of English articulation had some minor errors in it, which I tried to improve upon.
> If I pronounce la (laa) ... in English, my tongue ... at some point touches my gums behind my bottom teeth
That may be true for you, but this is not typical. Or at least, it's not the the "L" that's doing that. If you say "raaaaaaaaa" in English, you'll find that the "aaaaa" noise puts your tongue on the bottom of your mouth just as much as it does for "laaaaaaa".
And if you put your finger on your lower gums, you can learn to say "la lee loo lay low" without your tongue ever touching your finger, much less your gums. Just like how many English speakers curl their tongue to say English "ra ree roo ray roe" but they can learn to leave the tip of their tongue down (with practice).
I'm pretty sure I know what my tongue is doing in English, but alright it may be due to the vowel instead of the consonant. The problem in Japanese is there's no distinction between vowel and consonant.
Yes, it could be that touching the bottom of the mouth during the vowel part is not as important as the position of the tongue initially. l: relaxed, some sliding, top of mouth; l/r: tighter, as for a rolled r, more forward, top of mouth; r: does not touch top of mouth. loo and roo are both incorrect pronunciations of ru.
The more complete description of the Japanese r is that it is a tapped alveolar consonant that can range between a completely central [ɾ] and the lateralized [ɺ], with different degrees of lateralization. All will be interpreted as the Japanese r, and speakers tend to use [ɾ] after vowels, with lateralization likely to creep in if the r comes at the beginning of an utterance or after an n.
I'm a native speaker of Korean by the way, and while at the abstract phonemic level we also have a single l/r sound represented as ㄹ in the Korean alphabet, we do distinguish [ɾ] and [l] between vowels. 아리 ari uses the tap [ɾ] and 알리 alli uses [l] because that's how double ㄹ is pronounced. It's a similar distribution to the tapped and trilled r's in Spanish, where trilled r's between vowels can be analyzed as double r's.
Also, our "r" is much thicker than in English. For that reason, our English accent resembles a bit that of Russians :-)