Will it go into gear when the engine is off? If it does, then your tranny is likely OK. When it doesn't go into gear when the engine is running, that's an indication that the clutch won't disengage enough to disconnect power to the tranny and it's still being turned. When it happens because it gets really cold, that's an indication that there is air in the clutch system. Air contracts a lot when the temperature gets low. This, in turn, makes it seem like you're only partially depressing the clutch. Solution is to get the air out. Water in the system can also convert to air (vapour). If you have air, you likely also have water. Get the clutch fluid changed and bled.
If you are uncertain about the oil in terms of viscosity, then change it. Once you change it, there's no need to analyze it. The old oil is gone, so who really cares? If you are confident of the viscosity of the oil and when it was changed last and it's still got time and miles left, don't worry about whether it's synthetic or not. Just use whatever correct viscosity oil you have or wish to buy and use it for topping up. After the next change, you'll know exactly what you've got in there. Don't worry about mixing viscosities or synthetic with non-synthetic. This harms nothing and has been done as long as synthetics have existed. Ever heard of synthetic "blend"? If you have 4.5 qts of 5W40 and add 1 qt. of 10W30, you simply end up with a crankcase of a mix. Rough calculation would make it a 5.2W33 oil. No big deal.