Does downloading big files reduce my SSD lifetime significantly?
No, not at all. People underestimate the lifetime of SSDs. They can write a lot (usually around 700+ TB before clunking out).

This is a good place to find out more about how long SSDs actually last..

In fact, if you'd like to see more stats on your SSD, this tells you how to install smartmontools to get an idea of your drive's status.

As for your second question, I would say that's entirely up to you. If you'd like to avoid having large files cluttering your limited SSD space then go ahead with the HDD purchase.