As much as I'd like to see some changes to lissandra myself, here are my thoughts on your ideas
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using her ultimate on an allied champion isn't really what you want to do as lissandra. Using a really strong engage/pick tool to prevent an ally's death doesn't look like it's going to benefit her in any way as a strong engage midlaner. Imagine a strong lockdown like malzahar ult being used to make an ally in zhonya's state for the duration, it's just not as rewarding and I personally don't think it would help her.
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the recast while rooted seems pretty nice, but it's kind of stupid that this ability will completely prevent champions like lux from landing a full combo on you, and breaking matchups like this with a change like that doesn't seem very healthy for the game. Also, the main escape of this ability is for ganks in lane, as you'll usually only be in an unsafe situation after you've engaged and already used your e. As for the ganks in lane, only a few midlaners have roots, and it's already an excellent escape tool in lane that in my opinion doesn't need a change in that area.
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Actually quite like this idea
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There are a few problems with this added to her passive as A) in early levels you're simply never going to slow someone for 3 consecutive seconds as your q cooldown is too long, costs too much mana, and is your only slow, and B) even though this makes a good ult and the hit of consecutive Q's more rewarding, it could be too impactful, simply adding hard cc to a champion that already doesn't have too much trouble locking someone down, and definately needs the wording 'doesn't apply from slow from items'. It's like a kennen passive that decays quickly, but on a champion that already has built-in CC on 3 of her abilities.
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This oesn't really fit lissandra's playstyle as you dont' have access to a lot of different combo's with this, q already has a relatively low cooldown, and this doesn't synergize very well with her high mana costs early. Also, giving a 24 second ability cooldown (that has this cooldown for a reason) the potential of being cast twice in quick succession, will just make her too forgiving, and since this is the ability you'll most likely cast to start an all-in. Basically, you'll get a Q refresh for the most part, which isn't an ability you want to spam too hard early game as it consumes over 20% of your mana, and an E refresh on all-ins and sometimes to escape from ganks, which isn't something lissandra struggles with at the moment.
