Manual Trading on MT4 - Do I need a VPS?

The heading says it all.

I trade manually on MT4 - no EAs or bots or anything. Is there any reason I would need to run MT4 on a VPS?

I'm pretty sure (please correct me if I'm wrong) but if I put in a pending order with a stop and take profit, all that info is sitting on my brokers server. So when I close down MT4 and my computer is offline it will still get executed at those prices. Correct?

So I don't need a VPS then? Is it only for bots etc?

You don't need a running metatrader if you set tp/sl and manually trade.
You need metatrader running if you run the trailing stop option in metatrader.

suppose I have metatrader running for the trailing stop.

if the connection fails but comes back on later, will the trailing stop catch up?

If price has moved so that the price is further away from your stop, it will catch up to an extent , I believe.
I think that if, for example you have a 50 pip trailing stopand price has moved up 100 pips and then retraced 20, the trailing stop will calculate from what the price is at the time of the connection, not the high.
I'm not 100% sure about this though.