Is it foolish to buy 100 units of stock with $10 of processing fee?

You can take a look at the percentage of the transaction fees to the total price you are paying.

In this case you're paying $0.80/share x 100 = $80

Transaction fee = $10

Effectively, you're paying $80+$10 for 100 shares. i.e:

Cost per share = $0.90

It means you will need to see the share increase $0.10 before you break even. That means 12.5% increase. After which, you will see a gain. But anything less you are losing.

So to answer your question, if you think the stock can increase 12.5% (or $0.10 per share in this case) when you exit, then nope it's not foolish!