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I'm not an expert in this stuff, but I think there are a number of concepts getting jumbled up here. So a few thoughts:
1. You can roll an orphaned 401k into an IRA at any time with no penalties - no waiting period needed. You'll want to keep the structure the same (so traditional 401k --> traditional IRA and Roth 401k --> Roth IRA). If you're paying a 1.5% fee on your 401k, you should definitely do this ASAP, as that's eating into your gains significantly even if it seems like a small number.
2. Unless you're just letting it sit there as cash, all IRAs are "investment" accounts. You probably had some sort of default investment in your 401k like a target date fund, so that's why you might not feel like you were "investing," but you were. Likewise, in your new IRA account, you'll want to invest that as well. If you don't want to deal with figuring out individual stocks (which is the right move for most people), you can just pick another mutual fund (I recommend target date index funds if you have them available) to invest it all in.
3. I don't know much about Robinhood, but is there a reason why you're focusing on that one specifically? Robinhood is fine, but my perception is that they focus more on people who want to be active traders. If you're just wanting a place you can move it and forget it, I'd consider one of the traditional brokerages like Vanguard, Schwab, or Fidelity. They have an incentive to get you to move your accounts to them, so they're usually good at helping you with basic advice (not investing advice, but advice on accounts and how to roll things over) if you give them a call.
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