(1) The sound of ㄹ is not always [l] but the alveolar flap (like the sound of ‘t’ in the US pronunciation of water) when it occurs before a vowel. Thinking about how the tip of the tongue rapidly flaps in that case, I sort of see why ㄹ is shaped like that.
(2) I would say the claimed similarity between the ‘Phagspa script and Hangul is not really compelling. The shape of each letter in Hangul is so simple that you can just mention any alphabet system and I will be able to find a couple of letters that happen to look similar to Hangul. For instance, the Hebrew mem and nun looks quite similar to ㅁ and ㄴ with the same pronunciations. The similarity between ㄹ and Chinese 乙, which was used by Koreans even before the invention of Hangul to represent [l], is also notable. But these don’t really prove any connection between the different systems beyond doubt. Even within the ‘Phags-pa script, the letter for [ng] looks far closer to ㄹ than the letter for [l] is. And while the ‘Phags-pa script has a letter that looks exactly like ㅌ [t], in that script the letter represents [dz], which would be far closer to the sound of ㅈ. So why use similar letters for different sounds? Because the creator of Hangul cared about the relationship between the letter shape and the associated sound! If Hangul borrowed from the ‘Phags-pa, it probably chose what to borrow based on phonetic considerations.
(3) Even Gari Ledyard, the early proponent of the ‘Phags-pa theory, makes it clear that while the shapes of some basic characters may have been inspired by the Mongol script, the system has a whole was developed with phonetics in mind. His claim is only about what the initial inspirations are, and how they guided the development of the whole system.