1) You may want to check out whether an online-savings account (e.g. - Ally) or a no-penalty CD from such a bank would beat your current CD’s rate.
2) Ideally, you should have 3-6 months expenses available to you liquid. One month is kinda iffy.
3) I’d recommend against investing further in CDs until you have 3-6 months expenses available to you in liquid accounts. However, your plan to keep the checking account at $1,500 and start building savings sounds like a good one. Once you’ve hit your savings goal, if you then wanted to start investing in CDs again, you might want to look at setting up a CD ladder, so that even in your CDs, you’d be semi-liquid. Though, by that point, you may be better off looking at long-term, traditionally higher-yielding investments (equities, real estate).