There are separate boxes (eg Freeview [under 40 quid], or FreeSat [100 to 150] that can offer BBC iPlayer [probably via an ethernet cable to your broadband router, or if that's in a different room, can be done via PowerNet adaptors] however, I don't think any "off the shelf" box will have the same facilities, because presumably you could stream YouTube, and many other sites, without really thinking about it.
The run-of-the-mill TV boxes may have iPlayer but not a lot more. NannyNick has recently got himself a YouView box, but I've only a limited knowledge (from a few online articles) about what they can do, and while some of these boxes offer recording (onto a USB link Hard Drive) few offer facilities to record off TV as DIVX (indeed your Sony sounds interesting to me!!)
iPlayer on most boxes is not the same as on a PC - from what I've seen so far there's a menu of recent material but no A-Z whereas visiting the BBC site you'd have complete choice over what to view. A compromise might be to buy a similar size monitor, and feed it from one of the 'nettop' PCs so you'd have a wi-fi link to connect it to the internet (and with a wireless mouse and keyboard, be able to store it by the TV) plus some sort of recorder to time shift material either on Freeview or FreeSat (unless you went down the Sky subscription route)
As things stand, you can generally get SD or HD signals (standard definition, higher definition - HD best viewed on TV capable of displaying 1920 x 1080 pixels, aka 'Full HD' rather than some called 'HD Ready')
a) Freeview, Freeview+, Freeview HD boxes
some may include iPlayer facility
examples:
Freeview anything from 15/20 quid [but most TVs have Freeview built in]
Freeview+ - 250 GB, 320 GB or 500 GB hard drives. Can record and play only on that device (no way to move a recording to play on a PC)
Freeview HD - there's the iCan box (I paid about 30 quid at play.com and got the box and HDMI cable) which has iPlayer facility
see www.ebuyer.com/362680
b) FreeSat SD, FreeSat HD, FreeSat+ HD
FreeSat SD from 40 quid or so.
FreeSat HD from 65 to 130 or so.
If you have no dish at present, Tesco sell the Manhattan HD for 90 quid with free dish installation
FreeSat HD+ (recorder) from 120 quid (eg Grundig on Ebay) to 200++ for Grundig / Humax
Can view and record with the box, but again, unable to record for play on a PC.
Someone posted recently about using their old Panasonic (I will have to find the thread again) to record on hard drive and write to DVD. It may be possible to record signals on a SCART cable but HDMI is defined so as to be viewable on a TV but not recorded from an input (to limit chances of piracy, since signal is meant to be better than SCART signals)
Sorry if that looks like too many options, but you have the Freeview / FreeSat choice to make first (cover similar channels) then HDMI/SCART connection, and then whether to get a 'normal' 24"/ 26"/ 32" TV with Freeview in, which means you could use the TV to watch a show while recording on either Freeview or FreeSat.
Some TVs now include PVR (personal video recorder) by way of a USB socket, so you add a USB stick (or external powered hard drive for greater capacity) and let the TV record onto that. However I doubt many TVs have 2 tuners, so it would be recording what you were watching (or recording while TV on standby, just taking the incoming Freeview signal and storing to USB port).
Finally for more choice, if you had something like a Sony Playstation (PS3) you could view LoveFilm or Netflix or use iPlayer, and some other 'on demand' services from Channel 4 and so on. Only mention it in passing but there are quite a lot of options now.
Still love the idea of yours, where you could record as DIVX, though !