I think I might have the answer. It's a general rule. ę is not a palatalizing vowel, so the m in wezmę stays 'hard'. In weźmiesz, weźmiecie... the m is palatalized by the ie, so it becomes soft. And soft consonants in turn palatalize the sounds z and s to ź and ś, respectively. Maybe also some others, but I could't come up with any examples.
Example for z-ź:
Złe - źli (the i palatalizes the ł to l, and the l in turn palatalizes the z to a ź)
Example for s-ś:
Rosnę - rośniesz (the ie palatalizes the n to an ń/ni and that in turn palatalizes the s to a ś )
that makes sense, but why then it is weźmy and not wezmy (since y is hard?)
and in some other words it doesn't work either for example państwo (should be panstwo or pańśtwo then)
Good question. The singular form of weźmy is weź, which already has an ź. Weźmy is derived from that. You're applying the rule backwards now, saying that if there's a hard "my" it should change back into wez. But it apparently only works in one direction.
I said that the rule applies to z and s only. For n there is another rule that seems to be pretty consistent. When an n is followed by an s, the n is softened to an ń. Examples: Zieliński, Szymański, Jabłoński, ukraiński, prawdopodobieństwo, małżeństwo...
I hope that made sense
The stem is myśl. The infinitive derived from that is myśleć. Bot already contain the ś. Nowhere in the declension table does it change back to s. So, not really a problem. Like I explained before, it doesn't work in the opposite direction (hardening).
Rather look at verbs where it does change, like nieść or znaleźć and you will see that it works.
The only verb I found, which is a bit confusing in this respect, is wysłać. But here I would just say that it's irregular and the present tense uses a different stem (wyśl) than the past tense (wysł). This way I see no contradictions.
If you have further questions feel free to message me.
I don't think there's a better answer than "it just is this way". I guess it would be logical, given the fact that there's "weźmiesz" "weźmiecie" etc., but there's no Ź in the first person singular.