There is a tendency to lump all three sounds you mentioned together, but I would say it is not right. They say that ㅚ used to be different from ㅞ, but in modern Korean this distinction seems to be completely lost, so it's probably not worthwhile to try to distinguish between the two. They both sound like [we]. ㅚ and ㅙ are, however, pretty easily distinguishable. ㅚ is sounded with a small, rounded lips, not too different from when you say ㅗ. ㅙ on the other hand is pronounced with the mouth open bigger side to side, and the sound is created farther back near the throat than the frontal sound ㅗ and ㅚ. This difference is fairly similar to that between the vowel sounds of "when"([wen]) and "wag"([wæg]) in English. I think it's a good habit to try to consciously differentiate these different sounds because doing so will help us with our spelling too. Many people now get sloppy with this kind of small differences in pronunciation, to the point of treating ㅔ and ㅐ as same sounds, for example. The unfortunate result of this is that more and more people mix up ㅔ and ㅐ, ㅞ, ㅚ, and ㅙ, and the like even in their writing because they don't think about the distinction, proliferating spelling errors like 웬지(왠지), 근대(근데), and so on. So I think it's always good to be more precise with pronunciation, as you seem to be trying.

Here is a theoretical explanation. 1) 'ㅚ' ('ㅚ' is a monophthong). Locate the tongue position same to 'ㅔ'(middle front), don't move tongue and pronounce with rounded lips. Conventionally, it is allowed to pronounce as 'ㅞ' 2) 'ㅙ' Pronounce while moving tongue fast from 'ㅗ'(middle back) to 'ㅐ'(bottom front) 3) 'ㅞ' Pronounce while moving tongue fast from 'ㅜ'(top back) to 'ㅔ'(middle front) If you can pronounce 'ㅔ' and 'ㅐ' differently, you can also pronounce 'ㅞ' and 'ㅞ' differently. Even thought you pronounce them in the above way, most young generation of native Korean spkear may accept them all the same sound. Thus, you don't need to try to distinguish each sound.

Both ㅚ and ㅙ are pronounce absolutely the same. ㅞ is slightly different though. Fortunately, you won't come across ㅞ very often. They are mostly for mimetics and borrowed from foreign languages. e.g. 웩 (vomiting sound), 웨인(Wayne), etc. And there is no ㅗ+ㅔ.

yep you're right, 웨, 왜, and 외 too all have the same pronunciation. similar to the english words 'way', 'weigh', and 'whey'.