I've tried Lyumjev for a little while. Be aware it works much quicker than NovoRapid. For me that was very nice, as I needed to prebolus NovoRapid by 45 minutes or so to prevent after meal spikes, on Lyumjev I can just inject and eat. But if you're reacting quicker than me to insulin you want to watch out for hypo's, especially with meals which are slower to digest, like pizza and such.
I was already on Fiasp (NovoRapid's quicker sister, just like Lyumjev is to Humalog) when I asked for a couple of pens of Lyumjev, mostly out of curiosity.
It did unit for unit, so no issue there.
It also worked a tiny bit quicker than Fiasp for me (nice!) and it stung a little (not so nice).
The main reason I decided to stick with Fiasp was that I like my novopen5 with a memory function enough to not want to swap for the slightly better effect of the Lyumjev.
Had it been a choice between Lyumjev and NovoRapid I'd definitely choose Lyumjev, my life has gotten much easier with a quicker acting bolus insulin!
I've heard about Levemir and it sounds better than Lantus which I was previously on.
Levemir has a slightly shorter working profile than Lantus for most. It might be you'll find it's not doing enough in the hours before your next injection. Many people take Levemir split between morning and night, something to keep in mind should you notice you tend to go high before your next dose is due.