Although not ideal you can buy whats called 13 amp Trailing sockets, if you know how to fit a plug then you could fit these in place of the choc blocks.
Your new stuff including 4 way leads can then be simply plugged in, subject to load that you will limit at the supply plugtop fuse
or leave some money behind the bar!
Yes you need to confirm there is an earth core continuously from the plug via joint to sockets. You should have 3 cores within the flex brown(live/line), blue(neutral) and green/yellow (earth/CPC).
As already suggest the best method would have been a continuous length from sockets to TV appliances and a suitable number of gangs to the end of that. I would not use an extension lead on a reel, unless the reel was completed unwound and did not introduce a trip hazard.
There looks to be no issue with junction made but the flex and junction box should be securely fitted and not left unsupported.
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