What assurances do we have that we can safely store the DCR on evolution.dcrstats.com without risk of getting funds stolen, hacked, etc?
as usual, in this world, the words & reputation of the founders/developers of the project

so basically - none
Correct, but there is one more feature for security. All funds locked in tickets (PoS mining), so even if hacker get access to the server, he cannot steal any funds or he can get very little spendable amount (like 1-2% of total).