I have historically gone with PostgreSQL and had no complaints. The licensing issues concerning MySQL also give one pause (Oracle are greedy bastards who will use any excuse to extract money from captive customers, so depending on their properties is to be avoided). Having said that, these days, SQLite is probably sufficient for many workloads and has the advantage of not requiring a database server.

Free as in puppy

Drew DeVault recently wrote a simple but functional UNIX kernel in a new systems programming language named Hare in about a month, which suggests that doing something similar in Rust would be equally feasible. One or two motivated individuals could get something up which is semi-useful (runs on a common x86 PC, has a console, a filesystem, functional if not necessarily high-performance scheduling and enough of the POSIX API to compile userspace programs for), upon which, what remained would be a lot of finishing work (device drivers, networking, and such), though not all of it necessary for all users. Doing this and keeping the goal of making it a drop-in replacement for the Linux kernel (as in, you can have both and select the one you boot into in your GRUB menu; eventually the new one will do enough well enough to replace Linux) sounds entirely feasible, and a new kernel codebase, implemented in a more structured, safer language sounds like it could deliver a good value proposition over the incumbent.