connected with crossover cable
A crossover cable should work between any two computer (or like devices). Most often today, a straight cable also works, due to the near ubiquity of Auto MDI-X.
If the link comes up on plugging in you're good to go.
Static IP address: 192.168.0.1 Subnet mask: 255.255.255.0 connected via onboard ethernet port [...] Computer B Client: Windows 10 home Static IP address: 192.168.0.3 Subnet mask: 255.255.255.0
That's a valid IP configuration.
"B" pinging "A" or vice versa results in "request timed out"
If you haven't configured a firewall permission for ICMP echo that is to be expected. By default, ICMP echo isn't permitted on Windows.
Mysql cannot connect since both computers don't seem to see each other
If mysql is actually listing on that port (the standard port is 3306) and you've whitelisted it correctly then it should be working. Whether
ping works is irrelevant.
What is relevant is that ARP is working. You can check that by running
arp -a right after a connection or ping attempt. If ARP isn't working then either the link or your IP config is faulty. Then, if mysql doesn't connect either the firewall config or the mysql config is faulty - run
netstat -aon to check the latter.