Will my rear wheel with an internally-geared hub shift forward when pedalling steep uphill?
Wheels in horizontal dropouts have been reported to shift. You may or may not have a problem depending on how strong you are and how well your wheel clamps in your frame.

There are devices that fit in horizontal dropouts that hold the wheel in place and remove the need for re-adjustment after removing the wheel, such as Surly Tugnuts (example only, not a product recommendation).

There is no need to use a chain tensioner if you have dropouts suitable for adjusting the chain tension. It will only add friction and will not stop nor deal with the the wheel moving in the dropouts. 

Worn chains can not slip on non-derailleur drivetrains. A worn chain only elongates by 0.5-0.75%. When your chain becomes worn you should replace it as it will accelerate wear on the sprocket and chainring, as we'll as becoming inefficient.