People trying to log into my steam account every day

You technically can't do anything against this. I can try to access your account myself if I really wanted to and you would get a message. It doesn't mean I know your password or will ever access your account.

There is really no reason to change your password again, because the person trying to access your account is just using a script to run through a bunch of popular passwords probably.

Steam will automatically block the IP address after so many attempts to login to one account. They will need to contact Steam Customer Support to unblock their IP address and they obviously won't do that.

"Steam Account Password Reset" email

If you're received "Steam Account Password Reset" e-mail - you're safe. The only thing you can do is filter those e-mails so they will not bother you ("skip inbox" action in gmail filters).

"Your Steam account: Access from new device" email

Bad. Someone just tried to login with correct password. I'm not sure if it's possible to trigger this e-mail without correct password. You should contact steam support.

p.s. Some antivirus companies provide free "one-time virus scanner" (NOT a trial version!) like fully portable dr.web cureit (don't check "I'd like to receive news e-mails"!). Also livecd/liveusb version.