Will zombies follow me through fast travel

I really don't see how you can beat Breezehome in Whiterun for the combination of features and time.

While there is a loading screen, it's extremely fast even on my slow laptop (which is below the minimum specs for Skyrim in several ways) and it's just a few steps from the fast travel location.

There are two sacks on the shelf right by the door you can reach without moving once inside.

There are plenty of other containers for sorted, more long term storage. You can get the "Well Rested" buff instead of just "Rested" since you actually own the bed.

It's definitely the most convenient of the purchasable houses.

Save the game, press 'T' to check the day and time, fast travel to the place, press 'T' again to see how many days have passed. That will tell you how long the fast travel time is between those two spots. Re-load the save if you aren't happy with your purchase.