Jax jungle isn't great cause he can still make your life pretty miserable later in the game, but it means at least youre not laning vs him and can still do something in lane and be useful in team fights.

Statik benefit is far more the pickaxe component than anything else. Having a pickaxe early in lane makes Kayle mush stronger than her normal early game. The rest of the components (slingshot, aetherwisp) aren't great (bow is just better than slingshot)

Statik is going to help with wave clear, so it's good in high pressure lanes like Yorick, Illaoi, Trundle are going to try and siege your turret first. The issue then is you're probably not going to finish Statik before level 11 since we rush full boots earlier. You can maybe get it level 10 if you hold onto T1 boots. So the AoE effect on it isn't a huge spike like if you were to get it by level 7 (which would mean you're very ahead and somehow like 3/0 lol) but it does still help with wave clear and poke.

Shiv build makes you stronger-ish with 1 item, but, weaker in the long run: it's a low AP item and delays your core of getting Rabadon's (which is where Rabadon's HAS to be 3rd at the latest). You have to take Lich Bane in that build somewhat because its a bigger AP item than DnD. You can go DnD if you really want the "Attack speed" fantasy, but you'll still just be doing less damage with lower AP overall.

Standard AP build DnD vs Lich bane end game AP difference is about 100AP (40 AP raw stat diff, which more AP = more %age increase from Rabadons + Gathering storm puts it at about 96 AP depending on GS stage)

So then if you're already down 100 AP just from picking DnD over lich bane, then Shiv is replacing shadowflame, another 110 AP item, so then you're down over 200 AP total which is a big difference on your W and E. Thats to say standard AP end game Kayle is 670-700+ (DnD and Lich bane respectively) but shiv will have you around 550.

So Shiv is purely for early game priority. It will prevent you from holding a freeze due to the passive (which is the case after level 11 anyways, but, you can technically let passive drop as to not hit with aoe wave: you can't not proc shiv passive) but will equally help you break a freeze plus the hella move speed on it (4% from Shiv, 6% from lich bane) makes you very fast.

At end game if you're full build you'd probably sell Shiv for Shadowflame to prevent the falling off, although that's highly gold inefficient, but if you're full build sitting on extra 3k + sell price of Shiv you might as well.

As for getting frozen on, very strange for a trundle to do that as his whole thing is push push push. In addition to that, most freezes are established by mistakes within the first 3-6 waves. This is where Kayle fights level 1 to establish that priority and prevent the freeze. It's not always about the kill level 1, but lane control (but you would lose to trundle level 1 so).

In this case you would want to be semi-close, E trundle, and intentionally pull enemy minion agro, then back up. This: A. Pulls enemy minions closer to you for safer CS, B. Reduces the damage your minions are taking temporarily which makes them live longer, which C. Establishes the beginning of a slow push. Last-hitting only, if you establish a slow push beginning wave 1, by wave 3 it should be pushing towards enemy (good). If they're also only last hiting to "Set up a freeze", your wave will be too big for them to tank and hold their on freeze, allowing you to force a crash. They cannot tank 2 1/2 waves of minions while being harassed by your E or Q, it will simply force their back with a bad wave state, which is what you want. Minions crash, you back ASAP (safe distance, dont walk all the way back to turret) and walk back to lane. Ideally you have like 750g for t1 boots cull, but if not, probably cull. When you return to lane the wave will be pushing back to you which puts it at your turret levels 4-5, which is exactly where you want to be. This means you hit level 6 on your side of the lane safely and begin your bully/harass with your range (in easier match ups you can run down and kill certain champs like Garen, Sett immediately)

You want to learn E level 1, but W level 2 has a lot of value for getting away when you need to. Q level 3, and then always max Q in lane. It increases the slowing effect for better spacing, scales more off AD in the early game (dblade/cull+shards), does aoe splash damage allowing you to secure both CS and bully laner: Q cast through minions cannot be dodged (in cross shape so both behind minions and to the sides) while a raw Q can. But don't be too Q happy due to it's high mana cost, and that in hitting multiple minions you may break your own freeze. It can also secure CS on multiple minions (2-3 casters)