Can PEX be installed in sections?
You can do it a bit at a time.
Of course, any bits that are leaking that you don't replace will continue to leak until you replace them.
You can (or could when I was shopping, it was some years back) beat the price of a manifold with a handful of Tees and Valves. Pex does not require the use of a manifold, though the idea of using individual home runs of relatively inexpensive tubing rather than branch lines  splitting up to feed various fixtures may still have merit.
Where possible, try to treat PEX as PEX, not as rigid pipe. For very tight bends where you have no other way, you might need an elbow. Anywhere you have space, just bend the tube - it's cheaper and it flows better and it's one less spot to leak. Know and respect the minimum bend radius of the tubing you use (look up the manufacturer's specifications.)
In my local area, the piping within 3 (linear pipe) feet of the water heater has to be metallic (one of the changes they make to the IPC code which they base the local code off of.) You may not have that or a similar requirement - if you do, and if your water is eating copper, you might want to use stainless steel, which seems to be easy and affordable on the internet but gets the blank stare response at the hardware store. I would not suggest using galvanized iron pipe, based on every galvanized iron water pipe I have ever disassembled.