I'm not sure what you mean about 'working fine'. Sure, you can prepare a spell without the cold descriptor with this feat but it does nothing - it does not affect the spell.
This feat requires both the cold descriptor and for it to deal cold damage to function. Note that many abilities that change damage type do not change the spell descriptor.
Searching on these messageboards using this thread's title, most people seem to be of the opinion that you can prepare a Rime non-[Cold] spell and use Admixture to change it to [Cold] so that the Rime Spell works. I have also seen the opinion that it wouldn't work because the spell doesn't gain the [Cold] descriptor until you cast it and apply the Admixture ability (although note that Admixture says that you use it as you cast the spell, not on the tail end); this has the weird effect that your own Rime Spell preparation wouldn't work, but a Rime Spell Metamagic Rod would, and potentially has another weird effect that you could prepare an actual [Cold] spell with Rime Spell and then use Admixture on it and get Entangling Acid, Electricity, or Fire. I can see the argument in favor of the latter, but the former seems to work more logically and be the majority viewpoint.
I agree with you that it wouldn't work the other way around.
As I understand the Entangling Effect applies once the spell hits and takes effect rather than when it was cast or before it was cast. So if a cone of cold (acid) hits the enemy, it no longer has the cold descriptor and therefore no entanglement.
It cannot be prepared as the spell cannot normally be affected by the metamagic feat.
The workaround in most cases is the combination of the feats Heighten Spell and Preferred Spell to be able to cast said spell spontaneously, changing its descriptor and applying rime to it at the same time, as it then qualifies.