Engine makes noise when starting and shutting off for a short time

It could be the transfer case that engages the front drive, the drive shaft universal joints to the front, the front differential, or the self-locking hubs. The hard part is to figure out where the sound is coming from because it only will make the noise when driving.
It would be really unsafe to jack up the whole vehicle to run it without wheels on the ground. Probably best to take it into a repair place that will give you a quote, they may accurately tell you what's wrong. you don't have to have it fixed there.

If you are trying to run it on dry pavement, then something will be dragging although it's unlikely you'll hear it. Virtually all part time 4 wheel drive systems (not to be confused with all wheel drive or full time 4 wheel drive) make the front wheels turn just a bit faster than the rear. You will also notice that the vehicle jerks going around short corners like at a stop sign. In both cases the wheels are fighting each other trying to go at different speeds.

Next time it happens, hit the starter with a hammer or rod. If it starts, the starter is bad. Could check grounds also. My guess is the starter. Especially if it sounds funny or drags( goes slow)

what kind of car? need more info.

It sounds as if the belt is dry. Belts dry out after a certain amount of time and then start to make chirping noises. If the belt is not badly cracked, you can try using a belt lubricant spray. These sprays usually only last for a short period of time, though.

The whining noise at start up is either the power steering pump (possibly low fluid, or faulty pump bearing), or possibly a belt on the engine. As for the knocking sound, that is hard to diagnose without actually hearing it, but based on the fact that you are hearing a whining sound at start up, which is indicative of a loose belt, I would start at the belt tensioner and make sure that it is applying the correct amount of pressure on the belts. If the belt tensioner is faulty, it can make a knocking sound. If that is not the issue, then it could simply be the lifters. With engine run time/miles on the engine, internal engine components, such as lifters, valves, bearings and any moving part, deteriorate from friction wear and heat.They have a tendency to make a knocking/rattling sound when they are building up pressure. When they build sufficient pressure, or the engine reaches operating temperature, this sound/noise will dissipate and become nonexistent. This can be remedied by changing the oil, and possibly the weight of the oil (5w30, 10w40, 20w50) depending on your driving habits , or conditions (Highway v/s City/Stop&Go, or short trips under 30 miles)

I hope that you have found this information to be useful, and if I can be of any further assistance, please let me know. Thank you for using fixya!

Possibly timing chain tensioner.

try to clean the fuse box and keep it dry from humidity

This sounds like clutch interference. Sometimes the clutch breaks and causes a drag, but it all becomes "one" when the A/C in engaged. I suggest a close inspection of the clutch area. There is a release spec, for clearance, but this adjustment should not change, over time. All that matters is total release, without anything dragging.... Bet you have something in that area.

the blowing noise is coming from the fan clutch, best way to diagnose is to remove drive belt and start engine (only for a very short time) if noise goes away , replace fan clutch and retest.

Sounds like your oil pump inside the engine is not pumping oil fast enough when you start your vehicle its common problem for mitsubishi motors my car does the same thing.

good luck