To get 24vdc buy yerself a small psu
uk.rs-online.com
And an inline fuse so yer dun't blow yerself up.
To test a relay yer can measure the coil resistance while in circuit. Uvver components in parallel will alter the measured value burrits only an indication ovvit being open circuit or not. So it be ok. When taken out yer can power the coil and measure resistance across the contacts. The 'normally open' will be short circuit to the 'common', when powered. But relay contacts fail wiv pitting and carbon marks if swiching a lot ov power or have hadda lorra use.. So resistance ain't a good enuff test to find a volt drop that will only show when passing several amps across the contacts.
Removing components wiv a manual solder pump ain't easy. Especially if the pcb oles are tight on the pins and its a chuncky relay which soaks in the heat. Easiler to cut the track to the coil so yer can energise the relay in circuit wiv out removing it.
Scratch the track on both sides ov said cut and blob some solder accrossit to rejoin again.
To replace caps yer need the correct capacitance, volts and type. It should say on the side. Like 10nF 50v. Type yer know visually like electrolittic or tanterlum etc. Both spelt wrong assits fred rules spellin dun't count. Some caps are polarised.
Its a lorra faffing about just to save 200 sovs. But it will give us many weeks ov laffs.
If the pcb receives a signal from the main controller to switch said relay, why not just poke yer own version ov said signal in, and measure the relay contacts via the power connecters. Be easier.