Can things be unlocked in seeded runs?
Solution 1:
If you enter a seed, you will not be able to earn achievements or unlock items during that run in most cases but some seeds such as "qwop qwop" will still enable achievements while being a consistent seed. There is also no donation machine in the shop (but shop upgrades from previous runs still make more items available).
This is shown by the no achievement icon:
Solution 2:
Nope, seeded runs don't unlock anything. They also do not have a donation machine available, so you can't donate (or steal money from the donation machine by blowing it up).