Is there such a high chance a solution of LAH in THF is going to decompose so quickly (or in a worse case, self-ignite) if in contact with air all the time?
Yes, it can catch fire readily, I have seen at least one roaring LAH blaze, which was due to someone adding it as a solid but not under nitrogen. Not the way to start the day.
The key to LAH reactions is to transfer the LAH solution to the dry flask first (use a syringe or cannula), under inert atm., and then add the substrate dropwise from an addition funnel, which allows you to control the addition rate well. In all but a few cases, it is better to have the substrate in an excess of LAH from the start. You might even need to to cool the LAH soln. before adding the substrate, depending on the compound and the scale. Once the substrate is all added, let the reaction go to completion, then quench the remaining LAH with EtOAc (dissolved in THF, added dropwise also.), then do the Fieser workup or a variation of it (if not sure, google it). If the scale is over a few grams of LAH, then fire is a more serious risk, and some experienced help is best.