Personally I'd fix the pipework if it's in a bad state and the hoses aren't current standard with isolator taps and put both the washing machine and the dishwasher onto the cold tap. The chances are the plumbers aren't interested because it's a small job that whilst it might only take them 15-30 minutes involves a lot of heavy moving (washing machines are not fun to plumb in), plus travel time.
Theoretically this is the sort of job that should take possibly an hour or so, isolate the incoming water at the main stop ****, drain the water from the pipes, measure up and cut out/unscrew a suitable section of pipe around the old outlets, put in a replacement section of blank pipe for the hot water, and a section with the two outlet valves at the right places on the cold.
It'll probably cost a length of copper pipe, a clip on pipe cutter (about a tenner), and either a compression joint at each end of the new section or solder joint* along with the valves. i'd go with compression fit T valves (screwfix do them at ~2,50 and £6 depending on brand) rather than the "clamp on" type.
Total cost assuming you've not got any of the tools probably 50-60, and an hour or two of your time, if you've got a halfway decent tool set it's basically the cost of a bit of pipe and the fittings (and maybe a new hose to the appliance).
However, and I cannot stress this enough, if you do it yourself make sure you aim to start early in the day, and not on a Sunday - plumbing jobs tend to either go quickly and well, or you find that you need something you'd not planned on, and if you reuse an existing hose get some of the rubber washers to fit it, the old ones can/do go hard and may not be watertight if reused (you can usually get them in packs very cheaply), and a roll of PTFE tape.
It's also worth seeing if any of your neighbours you're on good terms with, or friends have any experience and would be willing to assist if needed if it's your first time
(watching youtube videos is helpful but having a second pair of hands to grab the bucket and hold the torch is always good).
It is if the hot water has been ran, otherwise you just get the first 10l of water that has said cooling in the pipework!
Yup
Also there may be specific reasons for wanting a colder feed, for one thing it means that the machine is able to regulate the heat better, with domestic hot water you could see the water out of the tap at 60-70c when you're trying to do a wash that might only want/need 30-40c (only pan wash programs tend to go hotter these days), and it can be detrimental for some items to be washed at the higher temperatures as some plastics for example can be damaged if you wash them too hot repeatedly.
Most dishwashers these days only need to heat a few litres up for their cycle, and usually do the initial rinse as a cold one.
*If using solder joints remember to thoroughly clean both sides of the joint where it's going to be soldered with wirewool.
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