I read on a website that the carbon in filter pads will remove the plant fertilizers you add to the tank.
yes, that's true. instead we use a biological Porous medium that hosts bacteria and microorganisms that converts nitrites to nitrates, and most of use also use Seachem Purigen, which removes other toxins and elements plants do not use.
I also read on a website that when you remove your carbon pads from your filter, you can't put them back in - wet carbon that dries out does not clean the tank anymore.
I've never heard of this assertion. I use larger more Porous Matrix Carbon that I keep in a 800 micron filter bag, and I reuse it anytime I have to remove medicine or tannins from my tank water. It seems to be effective even after being dried out for Months. Maybe on the more common smooth low surface area carbon gets "sealed up" by grime, reducing it's effectiveness, or maybe it's just a marketing ploy by filter media makers to get you to buy more of their product. Hopefully someone more knowledgeable on this point will chime into your thread.
Following this information, I can either have a carbon filter on my tank and no ferts, or I can use ferts and have no filter.
This is a contradiction all planted tank keepers must face early in their careers. The solution is to be creative with the filter you have, and not necessarily use the media system it's designed for. with a hanging filter on my 10g I removed all the media it comes with, and replaced it with small Porous stone media (biological filter) wrapped in a blue/white filter pad (physical filter) and dropped in a Purigen pack (chemical filter) after the filter area, where the water accumulates and pours out. In the long run using your own media system is also a lot cheaper since most of this can be rinsed out, recharged, and reused for many Months before requiring replacement.