It's a long time now since I used insulin pens but I always kept the pen I was using out of the fridge, I believe it is safe to keep it out of the fridge for up to one month, even after that it is ok but it is not so strong
Thats what I do, it does not need to be keep in the fridge when you are using it - well unless you're in 30+ degrees temperature in which case you do need to keep it at below 30
Most insulins should be fine outside a fridge for upto 30 days (which the exception of Fiasp which many think only lasts for about 2 weeks).
When at work many wouldn't be able to keep it is a fridge anyway (or feel safe keeping it in a work fridge)
Depends on the climate in your area. Below 25-30 degrees Celsius it should be fine outside the fridge but anything above that I would personally refrigerate.
It is generally not advised to inject cold insulin as it apparently is more painful but I haven’t experienced anything of the sort.
Around 99% of the time I keep my insulin and all diabetes equipment at room temperature and I only refrigerate my insulin if it is unopened or if it is opened and I’m in a hot country