My engine makes a squeaking noise for the first 30 seconds after

Sounds like excess bearing clearance, when the engine sits overnight the oil between the bearings and the crankshaft is able to run back into the oil pan. When you first start it up it takes a few seconds for the oil pump to fill the spaces back up and you get some clatter.

what is the engine oil like ? cat clogged you would know at any speed. could be a dry engine pump dirty sump pickup etc you need to locate the sound 30 seconds is usually the amount of time it takes for the oil to move round the engine.

If your Mazda Tribute's engine has a rumbling noise when you start it, this is most likely from the drive belt. You will need to replace it.

Could be the fan belt

Its because your alternator is a 80 amp which is a lot of power drain ,some propriety belt sticky sprays will cure it for a while ,arrow chemicals used to make one 30 yrs ago in the Uk which was very good but aqs to sorting this one out then short of renewing the spring loaded tensioner their is not a lot you can other than to start engine and dont rev it let it idle for a minute and this will recharge the battery so stopping the strain on the belt and the screeching at the same time

Go to your local auto parts store and ask for fan belt conditioner. Or if they have it graphite

The tapping is normal at start up, and is caused by lack of oil at the top of engine. The sound goes away within the time frame you stated (30-40 seconds). You have nothing to worry about.

check brake fluid level.

master cylinder might be of interest, but i would check the fluid level first

anything could set the service engine soon light,,but the noise,,,did you bleed the power steering pump?if not turn car off turn wheel fully left and fully right about 30 times then check fluid level and top it off,then start car and do 30 times with car on see if noise goes away,,,,maybe you got something on the belt to make it wet and noisy?

Sounds like a hydraulic camshaft lifter that drains down overnight and takes 30 sec to pump up with oil