What the heads and block are from doesn't really matter all that much. The C2 block should be correct for '62. The heads were used on 292s as well.
For street use, bore the block only as much as necessary. 292 pistons are probably the most common and should be very easy to find.
The cam is tricky. Given the difficulty of cam swaps, it is one component that you want to get right from the beginning. The problem with a lopey cam is that it requires other complementary upgrades in order to work well. You will need more compression, more carb, shorter gearing. With a stock stroke and typical overbore of a 292, domed pistons will be required in order to make a lopey cam work. Compression simply will not be high enough without them. If you make a plan and stick to it, you can build the bottom end for it now and swap intakes and heads later, when money allows. Without these other upgrades, it will sound good but performance will probably be quite disappointing.
If domed pistons, better heads, better intake and better gearing are not an option due to cost, then a copy of the standard 1957 cam or an Isky E4 or something similar should work well. It will not have a lope but it will actually perform better, all else being equal.
The crank and rods are ok to reuse. http://www.ford-y-block.com/
has had rods available new. If they are currently available then it is the best way to go because the cost to refurbish used ones is very similar.
Dual exhaust is one of the best single upgrades you can make.
A 4 barrel is a good upgrade but may not be absolutely necessary depending on how you decide to build the engine. It's also about a $600 minimum expenditure.
The water pump and oil pump can be new or rebuilt. Timing chain and gears, absolutely, new.