Quests don't have time constraints until they start. Time constraints within a quest are usually indicated: for instance, no matter how long you take to get to the Tarkus dungeon, it won't blow up, but once you start the quest you have to finish it before leaving or it'll blow up.
There are some quests that require specific real-world timing: most holiday-related quests will have an extremely small or no chance to trigger unless your computer's clock is sufficiently close to the holiday in question. This does sometimes have an impact on other quests: you need to do the Event Whorizon quest, which is a Halloween-locked quest, in order to enable the ability to transform Syri into her hellhound form. You can use the cheat menu to manually force these to become available.
Crew capacity won't matter unless you're aiming to pack the ship full with every possible crewmate. The Sidewinder is the best ship in the game in almost every category, it has one crewmate exclusive to it, you get it as part of the story, and it has 10 crew slots, which is more than I've ever recruited at once. There are 24 crewmates that take up ship space, so if you want to collect every last one of them, you'll want to enable the cheat setting for infinite crew space.