Elemental cores work very differently from mana cores, which is why shields work very differently in cultivation.
In cultivation you're strength is compartmentalized into forming Mana Soul stars until you have enough to break through to the higher level.
For elements, if a mage is a perfected basic mage, that doesn't even remotely mean that they will be able to use the basic spell for the element they unlocked at its strongest form, that's not even close to being the case. In the same way, one has to slowly accumulate their cultivation before they break through.
Unlike cultivation where one can compartmentalize their power, the power of your field is always fixed no matter how you slice it.
Here's how it works, when a mage awakens magic they have a fixed mana reserve (the amount of mana their body can contain before having to cultivate it or being unable to channel anymore), however, they do not have full access to said mana reserves in full. In their mana core as a basic mage, they have access to at most 0.1% without any stars, Each star they awaken grants them a full percentage of their mana reserves and in that light, the first part of cultivation (Increasing your cultivation base) was founded in expanding your mana reserves.
The second part of cultivation is the same way, except this time you would be permanently increasing the power of your spells; each element's spell has a maximum output, instead of only 0.1% though it starts at 5% power with no prior cultivation done. Each spell is broken down into 5 stages, Lesser at 5% with no cultivation needed, Basic at 15% after cultivating and fusing 5 stars with your raw basic elemental core, advanced at 45% after fusing 15 stars into your ordinary basic elemental core, superior at 70% after fusing 30 stars into the advanced basic elemental core and perfect at 100% power after fusing 50 stars with the superior basic elemental core.
Once a mage has perfected his basic elemental core he will not be able to burst through to their intermediate elemental core for that element unless their mana core is at the intermediate level as it would mitigate the mental stress of overclocking their elemental core and bursting through the barrier that blocks the element from the intermediate level.
The third and final part of cultivation one needs to worry about would be the speed of both cultivating and casting spells, for the most part, unless special means are used your cultivation speed is fixed, however, the speed at which one channels and casts magic spells is dependent entirely on precision and focus. Within a mage's elemental core at each level is a series of patterned channels, the more of the elemental core you unlock (unlocking intermediate fire magic after becoming an intermediate mage for example) the more complex the channels are to fill, that being said once one can fill said channels they can cast the spell for the given level of said channel.
While it is very much possible to double cast it is complex and requires special conditions. There are four types of double casting, double casting different elements spells simultaneously, double casting the same spell twice simultaneously, double casting different spells within the same element simultaneously, and casting a spell on multiple targets.
Different spells, different elements: This requires a mage to first have multiple elements awakened so they will likely be an intermediate mage, They would then have to partition their mind towards channeling magic through both elemental cores, and they would then partition their mind once more to channel and equal amount of mana through both elemental cores simultaneously, they would then fork the spells mana channels simultaneously and ignite the mana within the core activating the spell. From there it's a matter of directing how the spell will be cast as the mage in question will feel the spell being cast through them.
Different spell, same element: You also have to be at least intermediate level to do this, unlike before where you partition your mind 3 times, you only do it twice in this instance, you do it once good twopence into the selected elemental core and start drawing the channels of the two spells you want to cast and ignite once done.
Same spell same element: the same as the prior one except you are drawing the same spell channel twice simultaneously.
Multiple targets, single spell: This is less partitioning your mind and more directing the spell as you ignite your mana.