Please find reply below:
I have chain lock around my grilles for added security, see photo attached. Can the insect screen still fit in or must I remove the chain lock?
No... Insect screen need to paste on the frame.
For the kitchen one, since we hang our clothes outside to sun everyday, would the daily removing/handling make it less stick / secure (from insects) / durable? Typically, under such circumstances, how long (in years) can the insect screen last?
If you are referring to magnetic strength, it is designed / made for removing / opening / closing... Durability depends on individual & how screens are handled. Typically it will easily last more than 5 years.
DIY Magnetic Insect Screen 2018-04-12