Price-wise I'm going to say £500 just to plant a flag in the ground and get the conversation going. But like all things, it's not the final word.
Ignoring the Chinese projectors, there are something like 130 different models under £1000 currently available from reputable projector brands. Although the lowest cost projectors with either 3xLCD or DLP light engines start from as little as £250 (e.g. Acer X118), you really don't want something like this for home entertainment.
Within the projector market, there's 'budget' and there's 'entry-level'; these two things are not the same. Budget products are the cheapest of the breed that a manufacturer can make. All the fat is cut away, and quite a bit of the lean, too. You end up with the lowest spec and lowest performance. These things are designed for occasional workplace use where no one is being critical of the image. They might be noisy, have low resolution (SVGA 4:3), maybe some image brightness issues at the edges, and definitely low fidelity when it comes to motion handling and scaling and colour accuracy.
For home entertainment or home cinema, and entry-level machine isn't the same as a budget projector. These are designed with video in mind. The light output is tuned for contrast, and the panel will invariably be 1920x1080 HD. The motion processing and scaling will do a fairly decent job of handling upscaling and movement. The fan noise will be quieter because, in most cases, the light output won't be screwed to the max for PowerPoint.
An Optoma HD27e is an entry-level home entertainment projector at £450 new that's worth owning. Around a similar price you also have the Benq W1050 and Epson EH-TW650.
These are still at the shallow end of the market. On paper, there won't appear to be that much difference between these and 1080p machines at £4000-£6000, but performance is another thing altogether. In fact, I could show you the Benq, then the Optoma, and you see an improvement, then show you a s/h JVC for £700-ish that would have been £3500-£4000 new and you'd be absolutely floored with the difference.
I wouldn't use a JVC for what you're planning. It's a big old Hector, so is far better suited to permanent installation; but the point really is about the steps up in performance. The cheapest proper machines aren't in the game for video at all. You'd be just as well with the £130 pj from Amazon. Even the higher-level business machines are relatively poor for video, and that includes product at £2000+. Doing video justice, even if its as a background, means getting something designed for the job.
