Will my rear wheel with an internally-geared hub shift forward when pedalling steep uphill? (I'm not a bicycle mechanic. Please excuse the incorrect terminology.)

I would like to have an internal hub gear (Shimano Alfine 11) installed on my bicycle. My bicycle has a horizontal rearward-facing track fork end. 



Situation 1: no chain tensioner

I was told that to keep the chain tense the rear wheel needs to be pull backwards and then the lock nuts tightened. 

I am concerned that on a steep uphill the rear wheel will shift forward, e.g. if I have a bad puncture and I need to replace the tube, then I need to make sure to tighten the lock nuts to prevent the wheel shifting forwards. 
Question: Is my concern valid or will the wheel never shift?

Assuming the chain is worn out and starts slipping and there is no new chain available. I could re-tense the chain by loosing the lock nuts and shifting the wheel back to tense the chain. Question: Is this correct?



Situation 2: with a chain tensioner

In this case, I could install the rear wheel right against the front of the fork end. The chain tensioner takes care of the tension. Question: Is this correct ?

Assume the chain is worn out and is slipping and there is no new chain. Question Can I adjust the chain tensioner to increase the tension on the chain, i.e. does the chain tensioner only "have one tension" or is it possible to adjust the tension of the chain tensioner?
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.