No reason why not unless it is subject to very low temperatures when a 0W/30 would probably be more suitable. 5W/30 is recommended for normal use in a temperate climate with increased oil changes for harsh conditions.
You may have air in the line. Check the level in the reservoir. If it is very low, you probably got air in the line and will have to bleed the system at the slave cylinder. Before doing that, fill the reservoir, pump the pedal, and check for leaks. Repair any leaks before bleeding the system.
It could be the "oil pressure switch" which on most vehicles.....is a small device that screws into the engine somewhere. It also has a wire connected to it that sends the signal to the light or the pressure gauge on the dash. It could be defective.....or just has a bad connection (could have an oil residue on the connector); Google it to find where it is located and see if there is an auto parts dealer who has a replacement sensor switch if you are looking for a possible way to isolate the trouble. Most of them sell for about ten dollars or less.....try O'Reilley Auto Parts to compare prices....Good Luck
The transmission shifting is controlled by computer, so if the gearbox has been replaced the problem with the shifting lies in the wiring or the Transmission control module, what that issue is will require special testing and likely a wiring harness pin out breakout box special tool. I would suggest you seek professional help on this, but that is entirely up to you.
