As a mechanic of 30 plus years, and full time tinkerer, I have run into these issues with pens from time to time, but me being me, had to figure out why, and how to fix it.
Firstly, 2 different things cause 2 completely different results. If you find your plunger hard to push, or impossible even, the most likely cause is the fact that the inner needle that punctures the rubber end in the vial simply isn't piercing completely into the vial. The most common cause of this is the fine thread style of needle being "cross threaded", and not actually screwing on far enough to pierce the rubber on the end of the vial. No hole, no insulin. If you cross thread the needle a bit, then take it off and start over properly, you might get lucky and it will work, OR, you may have bent the inner needle, and you have to toss it and get another one.
Another cause of this, when using the quick attach needles, is not turning them on fully until they click. Because of the nature of the coarse thread on these, a small turn equates to possibly a millimeter of penetration of the inner needle. Needle manufacturers keep these lengths to a minimum to save materials in manufacture, so they are designed no more than long enough.
As for the not moving the inner plunger, some needles I have had seem as though they bypass if you put too much pressure on the plunger. If you are pressing at an awkward angle, or simply pushing too hard, it feels like it just falls in, not administering a dose, which it is not. I found that a slow, easy pressure generally remedied this.
One VERY important note: NEVER use a set up designed for one brand with another, especially like if you changed from say, Lantus to Trujeo. The are completely different internally. Trujeo is 3:1 as strong as Lantus. If you put a vial of Trujeo in a non-Trujeo pen, you will dose with 3X the amount you intended. If ever in doubt, get your Rx changed to auto one piece pens instead of replacement vials. The Novo rapid pens are exceptional, as they are spring loaded internally, no long end coming out that is hard to use with one hand. Simply dial and put mild pressure on the end button, the spring pushes the plunger in for you.