New quests don’t appear

As of the patch that rebalanced armor, weapons, and enemy difficulty, this fight has become much easier. I completed it easily with one or two bandages, copper armor, an iron bow, and an iron melee weapon. The summoned penguins all died in one to two shots of my Hunting Knife. After that I just needed to keep firing arrows at the mothership and occasionally dodge its blast. Even if I got hit, bandages could easily keep up with the damage.

I'm pretty sure you don't even need to bother with copper armor to beat this in its current state. Unless you're absolutely terrible at combat, you probably need only the following to take out this boss:

• An iron weapon such as the Hunting Knife (10 iron bars, 30 pixels), to dispatch penguin enemies quickly.
• The iron bow (15 iron bars, 150 pixels, and your previous hunting bow) to shoot at the mothership. You could probably do it with the normal Hunting Bow too, but why make the fight more tedious on yourself?
• Bandages (4 plant fibre each). 10 should be plenty, but if you want to be sure, just stock up on a bunch.
• A basic shield (10 wood planks, 10 iron bars, 20 pixels) is by no means necessary, but binding it to the right mouse button can be handy so that you can block while bandaging yourself. You can use bandages while blocking, which means you can basically back yourself into a corner and become immune to damage (or at least take greatly reduced damage) while you heal.

Total resource cost to prepare for this fight (assuming you've already crafted everything along the quest line to get the ship to appear): 35 iron bars, 10 wood planks, 200 pixels, and 40+ plant fibres.

If you've got the copper handy and want to make yourself some copper armor, knock yourself out. I don't think you'll need it though. And if, like me, you had a ton of silver sitting around too, keep in mind that after this fight, you'll probably just be upgrading straight to silver armor anyway, since it unlocks soon after.

The only way to store pixels (as of yet) is via the Pixel Compressor.

Pixel Compressor is a crafting station used to turn pixels into tradeable/storable items called voxels. Decompressing these voxels will yield the equivalent amount of pixels back, it provides a way to trade pixels to others as well as protect pixel loss from death.

This crafting station is crafted at the Architect's Table (third tier Inventor's Table), and requires

• 15 x Gold Bars
• 15 x Durasteel Bars

You can only compress 1000, 2000, 5000 and 10000 pixels. To decompress them, simple use the decompress table (which acts much like the sell window on a merchant).