430 3rd Gear Dropping to Neutral

Hi, All. New to the F430 ownership and I have a 2007 coupe. I was wondering if someone can help me? No history on the car and finding out that the car will not upshift or downshift into 3rd gear and instead shifts to neutral. I have to upshift to get it back in gear and usually starts at 6th and I downshift from there. Happens after the car warms up and in auto or manual. Outside of that, the car shifts very well in all other gears with no slippage. No strange noises or leaks that I can see. Just doesn't like 3rd gear. The car seems to have been kept very well. Does anyone have a short list of things I can check before I head to the dealership? I am very mechanical and have worked a lot on a lot of foreign cars in the past. I have also disconnected the battery for 20 minutes, touched the cables and did what I thought was an ECU reset. No service lights on, either. Thanks for your help!

Photos of the car and engine to provide context on the level of care from the previous owner.
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So here are some answers:

Door Opening: 3.81 Seconds
Key: 7.65
Gear Shifts: The gear shifts from N to R and R to 1 and 1 to N seem pretty good/quick. Pump comes on for a ~3.8 seconds from N to 1.

Mileage is 43,100

And will the Autel allow things like measuring clutch life and bleeding the transmission? Or is it better to buy the Launch 431?

 

So the pump stays off for 14 minutes and 25 seconds before turning back on with the engine off. I waited 16 minutes and 32 seconds with it idling and couldn't hear it so I think I may have missed it. Nor could I hear the pump shifting when I was inside the car with the engine on and changing gears. The car is super loud with no muffler.

 

OK so @flash32 has been so helpful and if you are following this thread, here is where we are right now:

1. Purchased the ThinkDiag software which is the equivalent (and way less expensive) to the Launch 431 software. Absolutely love this product.
2. We have determined the clutch wear is at 39% (61% remaining).
3. F1 pump timing is a little borderline but passes specs.
4. Identified a P1770 code for gear engagements with wrong shifts.
5. We ran a relearn on the clutch and still awaiting a test drive.
6. Potential next steps include changing the accumulator and doing a full bleed, followed by clutch position sensor replacement and regulator valve replacement if needed.

Codes have all been cleared and heading out for a drive to see if anything has changed with the clutch relearn. Also running real time diagnostic reports during the drive.

Updating later....

 

@flash32 is really amazing. Thanks for all of your help walking me through the troubleshooting and it looks like my car is shifting perfectly now. Still going to change the accumulator, though, after driving for another 1-2 weeks. But the relearn and all of the diagnostics show my car is in good health and working properly for right now.

Could not have gotten this far without your help!!