I'm going through a similar dodgy hack at the moment.
A 7 speed deraillur may have enough physical range to cover the width of an 8 speed cassette, or it may not. Note an 8/9/10 speed cassette are all the same physical total width.
You will require a 10 speed shifter, and the main problem here is whether the cable-pull matches the cable pull of the 7 speed derailleur. Using a friction shifter should work, but that's an aquired skill and its not particularly ergonomic.
It is safer to fit a 10 speed rear derailleur, shifter, 10 speed chain, and cassette all at once and all from the same vintage/groupset.
Additionally, theres "dynasys" which is some Shimano marketting-droid's buzzword for "yet another incompatible system" so make sure your rear shifter and rear mech match, either both or neither.
Regarding the front, you can use pretty much any crank you like that fits the chainring. The chainring will probably work fine too, but if you ride bumpy things then a chainkeeper/guide would be a good idea. As a frugal solution you can use your existing front mech but use the limit screws to lock it into one position, and do away with the left-hand control. This will help keep the chain on the front chainring.
Later, if you wear out the chainring then consider buying a narrow-wide chainring, which should mostly negate the need for a chainguide.