Quick answers:
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Yes, Home Assistant is a standalone home server that doesn’t not need an outside service/account/subscription. It can simply collect data and analyze or you can use it for more complex automation tasks (I just use mine for data collection and some analysis). Home Assistant does collect data directly from local devices, but also from the cloud (i.e. Sense and Ecobee data are pulled from your account).
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Both - Home Assistant is accessible via a Web App and Phone/Tablet app, just like Sense. Both provide useful viewing and control, but configuring and editing are better done via the Web App.
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I just looked at specs, and you would really need to add a 128GB or greater NVMe solid-state drive (SSD) in Yellow’s M.2 slot. The amount of data Home Assistant keeps around is configureable - I keep around 30 days of data across Home Assistant and a couple other apps (influxDB and Graphana), and I stay safely below my 128G, but 64GB wouldn’t have been enough. I would recommend 256GB
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There used to be a nice Home Assistant Blue that came with 128GB, but they had supply chain issues and have been out of stock forever: