This could be a DIY project, but it's pretty big. Anytime you start messing with support columns you can get yourself into a big project. But, that said, this is most likely doable.
You need to get the bottom of the post off the ground with some sort of concrete footing and replace the bottom (or all) of the post.
Basically, if you hired a contractor, he'd prop up the roof using doubled 2x8's on each side of the post. Then he'd either replace the whole post or cut off the post about 2 feet from the ground. He'd probably (based on code) also dig a hole and pour a concrete footing and then bolt the new post base back in. The size and depth of the footing can vary by local regs and the contractor's preference.
If it were me, I'd prop up the porch, take out a post, dig a hole (2 foot+, unless you are concerned about frost heave in which case 4 feet+), put in a sono tube up to about a foot above ground, fill it with concrete and put a new post in.
Based on local code, this would probably require a permit (but depends on where you are and the nerve of your contractor - personally, I'd do this myself without a permit...)
You reuse the metal bracket by embedding in the top of the new footing (sand it and paint it with rust proof paint), but you might not need it. Home Depot has lots of various bracket to attach vertical columns to footings. Also, google images is your friend on how this can look.
You need to troubleshoot the problem and this is best done with an electrical portable meter. You probably don't have one, a local harware store would have it. Safety first in everything you do here. If you have fuses remove it from the bathroom circuit at the main panel.
The "lamps" in the bathroom need to be tested in another location and the wiring coming into the electrical boxes need to be inspected (this includes the switch box) and I recommend a new light fixture and/or a new light switch hardwired in the bathroom. Then replace the bathroom fuse at the main panel and flip the light switch on. Good luck.
Best Answer
Partial answer is if you don't have and don't want to purchase a reliable electric meter/voltage tester than you should hire an electrician, the meter is to ensure the circuit is truly dead off. Don't use a $20 cheapo multimeter on mains circuits they have poor fusing and are dangerous. Always begin by checking the meter is working by testing a live circuit.
If the wiring just comes through a plain hole in the siding without an outlet box then it was not installed properly, it will need a remodel-box installed, assuming the remainder of the wiring install is acceptable this is a simple matter of buying the right size for your lamp's mounting screws. If there is no box and the wire looks old, less than 6 inches of extra, undersized, or lacks a ground wire then hire an electrician.
If there is a box, is it the right size and style for the lamp base? It is somewhat odd to have it screwed right into the siding rather than the box, but not inherently wrong as a few are designed for that. If the screws pulled out due to wood rot then you have much bigger issues, if they were just undersized screws then you can put in some wall anchors of appropriate style and size.