Should you ever solo push a tower?

Yes. If your team is holding a lane, and all five of the enemy are there, its often worth to go splitpush. If your team is strong enough to hold off a 4v5, good for them, you get towers for free. If they aren't, splitpushing is still a pretty good idea, since it forces the enemy to send people after you.

If you see 5 people on the other side of the map, keep pushing. If you see nobody, you may be going to die soon and you should retreat and improve vision.

If you can verify that nobody will be able to sneak up on you, due to deep warding or distractions elsewhere on the map. Always start running as soon as someone is MIA and could be heading your way, unless you know you can fight them.

If you cant see them, they are everywhere until observed.

It all depends on vision and the situation.

Like I was playing a game as nasus yesterday, our team had baron and I told them to go top and i'll go bot since we had mid already because I was fed and I knew they would have to send multiple people to deal with me

I go for the second tower in bot lane and four of them showed up, and it took them so long to kill me my team took the top inhib and we won shortly after