IANA real plumber! (but I have done a bit of this...)
TRVs can be pretty cheap and nasty, but something sensible quality like a Drayton TRV4 is going to be around £25 per pair (TRV and lockshield valve), so around £125 all in. Even if you drain and re-fill the heating circuit yourself you'd be well advised to add Fernox or some other inhibitor when you re-fill, which is going to be around £20 more. Local plumbers around here charge from about £80/hour labour + VAT so I don't think your quote is too unreasonable as long as you trust their work. If you go down the plumber route, then make sure you know what sort of TRVs you're getting and consider whether spending an extra £50 might get you something nicer looking and reputable.
To drain the system you're going to need to slacken off the connection between the radiator and the valve and start catching water in a tray or bowl which will be a long an tedious job. You also have the option of using a freeze kit I guess but that'd work out adding a good few quid to the job. To ensure the whole system drain down, open up all the rad valves around the house first and open up a vent at the top of the system. On the radiators where you plan to install TRVs, it's probably worth taking the rad off the wall, taking it outside, and flushing it manually with the garden hosepipe before refitting it with the new TRVs. You could quickly whip a whole rad valve off the pipe and swiftly clip a length of suitable hose to the pipe using a jubilee clip I guess, and then run the house out the nearest door to drain the worst out of the system from a single point before using a bowl to get the dregs out of each radiator.
Given the 8mm microbore piping and its tendency to sludge up, the full chemical flush is probably not a bad plan to clean out the system.
Having just moved in, I guess you won't know how well the system works currently and whether or not the radiators all get warm properly or whether there are cold spots or ones which don't seem to work. If the latter, then the flush is probably money well spent.
When replacing TRVs, there is always the risk that the tails into the radiator will need changing and won't be the same length as the old ones so then you're into modifying the adjacent paintwork. If you're not a confident DIY'er then it may be best to pass all that risk onto the plumber, making sure he's got that covered in his fixed price.