Engine and oil lights on

Thinner oil in the winter time will help. 0w20.

It is most likely a worn out relay tthey when it's could metal contacts in relay are not making connections I know it sounds weird but metal changes major in different temps

Change the transmission fluid and filter. The light says fluid temp. is high, so you may have some burnt fluid-deadly for transmissions. Drop the trans. pan and look for metal shavings-too much will indicate failure of transmission parts. If not too much, you may be alright with fluid and filter change.
Stop the leak,. make sure it is trans. fluid, not motor oil. If motor oil, stop the leak and change oil and filter. I would do the transmission regardless, since the light came on.

They are real easy,here's a link on how to replace.Here's the switch.

giving the fact that your oil pressure drops tells me that its a high millage worn out motor when the motor is cold the oil is thicker then when the motor is at normal running temp. some people have good luck with using a thicker grade oil such as 10w 40 or 25w 50 and some will still run the same oil as they always have but just use a additive such as lucas oil treatment witch is thicker but works great for worn motors slick 50 is another really good additive as well both of witch i have used and my vote is on the lucas just because i run it in my stock cars we race and all my older trucks.

Measure your oil at normal operating temp, without the engine running of course. This will give you a more accurate reading once the oil has circulated in the system.

The system works like this:

• With the ignition on, the oil pressure indicator light on the dash should come on and flash. Once the engine is started, the light should go out and stay out as long as the engine is running.
• If the oil pressure drops below approximately 0.3 bar or 5 psi when the engine is running below 2000 rpm, the light will flash.
• If the oil pressure drops below approximately 1.8 bar or 27 psi when the engine is running above 2000 rpm for more than 3 seconds, the indicator light will flash and the buzzer will sound.

If either this two warnings occur:

• The engine should be switched off immediately and the oil level should be checked.
  

Could be the oil pressure sender has gone bad or worse case scenario the oil pump has worn out. Check the oil presssure sender first, if it is leaking around it then that is the cause, if not then you may be looking at an oil pump. Just had a 03 ram 2500 doing the same thing and it was the sender, but that was a hemi and it had 230000 miles and a month later had to replace the oil pump for same issue.

You should be using 10W30 in summer. You may have 5 W still from the winter.

Did the temp gause go up past 1/2 way?

The oil pressure warning light only comes on when oil pressure drops below 4psi at this moment there is no oil in the engine most often it does not get to this point because oil, water etc is checked every week. It could be your engine was using oil which would point to an engine fault.