Advice on getting from Eilat to Petra on our own?

We are planning to make a trip to Petra from Eilat on March 4th and 5th. Since the tour prices are very expensive, we would like to explore the option of going on our own.

I need advice as to where to cross the border, what taxi prices in dollars should we expect Eilat to border and border to Wadi Mousa. There will be three of us and I already found a tripple room in Wadi Mousa for $ 50.

What taxes should we expect to pay at the borders when entering and leaving Jordan? What is the opening time of the borders?

Is it possible to arrange a taxi driver to wait for us at the border? where do I get a reliable one?

Does the rate of JD50 for a day entry to Petra apply to foreigners and what 'services' are included? I didn't find that on their webpage.

My dad is ill and can only walk slowly and in easy terrain. Can we get him a horse and how much would it cost? can it go everywhere?

What would be the weather like in beginning of March?

Any other advice/experience you can offer would be great!

Sorry for too many questions :-)