To check manual transmission oil you need to remove the filler plug. The plug is a 24mm bolt on the side of the gearbox. On the lower side of the gearbox there is a 24mm plug which is the DRAIN.
Remove the filler bolt, and oil should run out of the gearbox, or at least be up to the hole. So you might need to stick your finger in the hole, to inspect it. To fill you might need a small pump or syringe to squirt some gear oil in.
From the problems you have, low oil will not cause the gear to 'pop out'.
You have a transmission movement problem, either internal, or external movement.
External issues can be linkage problems or mounting problems. Linkage problems are easy to diagnose and can typically be duplicated. Worn engine/transmission mounts can allow the assembly to move and cause strain on the linkages which can cause the pop-out. This is a little more difficult to find.
Internal issues are caused by a few items, but the most common one is worn bearings. The helical-cut gears used in production cars put great strain on the bearings. Helical gears are used because they run much more quietly than straight-cut gears. (However, helical gears are much more expensive to produce, so we know cost isn't the factor here.) The two shafts that are engaged with helical-cut gears are always trying to force each other apart along the axis of the shafts. This force is transferred to the bearings and then onto the case. Once the bearings begin to wear, the shafts develop a small amount of play and can then move along their axes. Since the engagement forks and their respective hubs also move in a motion along the axis of the shafts, any movement is going to effect gear engagement. The shift hubs are normally free to move along the shafts and once pushed or pulled into an engagement position they are held in place by only small "detent" springs. These springs need to apply only a minimal force to hold the hubs in place, but if the shaft was to move along its axis, such as in a "power on, power off" application, the detent springs have no chance of holding the hubs in place.
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