Would uploading a file, program, script, etc. to an online storage service, without sharing it, count as distributing the file to the online service?
To me, it depends on the terms and conditions of the remote storage service. If the service itself acquires rights to use content you store on it, then yes, it is very possible this constitutes distribution. But if they're simply providing you a safe repository wherein you store data, then it no more constitutes giving them a copy than putting your valuables in a safety deposit box constitutes giving them to the bank.
If I backup a python script not intended for distribution in a private GitHub/GitLab/etc. repo, which imports a GPL library, do I have to comply with GPL?
I don't intend to read their terms and conditions; if you decide to use their private repositories, that's your job. But assuming they don't acquire any rights over and above those needed to safely store a copy of your data and return it on demand, then by Q1 I don't think it does. If they do acquire some rights, then we'd need to know what those are, at which time apsillers' excellent point may come into relevance.
Normally, running a GPL web service is not distribution. However, what if the web service is run on a server like AWS, where you have to upload it first? Do you have to comply with the GPL for the upload?
I'd hold that AWS is rented space, like a safety deposit box, and thus Q1 still applies. I've stored quite a lot of data on some fairly expensive AWS deployments over the years, and I've never heard the slightest suggestion that that data belonged to anyone other whomsoever owned it at the time of storage.
If you make it available to anyone else from that storage, then distribution definitely occurs.