After speaking with the recruiter for the first screen, is it good practice to send a quick follow-up email to the recruiter saying "It was great speaking with you today. Looking forward to next steps!". Does this convey enthusiasm and focus or being a over-eager and insecure?
I wouldn't do this if the recruiter already conveyed next steps/timelines with you. (here's the more general discussion about thanking people after interviews.)
I'd use the "looking forward to next steps" line as a gentle reminder that you're still waiting for a reply if the alotted time has passed.
The way I view it: each interaction with the company should add new value or new information. I don't think your message passes the bar.
This feels unnecessary to me. I would only do this as a friendly ploy for re-engagement if the recruiter is ghosting you after the screen.
It's not exactly the same, but the spirit of this interaction feels similar to what I cover here: https://www.jointaro.com/course/ace-your-tech-interview-and-get-a-job-as-a-software-engineer/dont-be-weird/