Versaille Entrance

A partial answer--I visited in May 2016 and had a Museum Pass, went through the main entrance and dealt with the usual lines for entrance and then security. I ended up in the Trianons and Hamlet (my very favorite part of the whole day--peaceful, serene and I had it mostly to myself) and exited from the Queen's Gate and walked back to the train station from there, first a side street then I cut diagonally across the bottom end of the parking lot. Maybe a 15 minute walk. Access to the park grounds is free and no monitoring of people coming and going so no lines. I expect others will be able to address your other questions.

Answer: You around to the front and get in line.

There's no back entrance from the gardens.

the only trick to avoid the main security line is to get the King's apartments tour which has its own door. You can go to the gardens early through the gate to the right of the Chateau; it is usually just open; on fountain days it costs 8 or 9 Euro and there is a ticket booth. But you can't get into the Chateau form the garden; you have to go back out front and get in the line which has always been long when I have seen it; it is certainly hugely long in the morning; don't know if it gets short in the afternoon.

Thank you .

I have entered through the Queen's Gate many times on foot and via car. On foot or by bike there is no charge. Occasionally there are a few cars waiting to get in but pedestrians just walk through a side arch. There is a very small parking lot just outside the gate - you have to get there early to get a spot though if you end up with a car for some reason!
I would recommend either renting a bike somewhere in town and biking in through that gate (I have no idea where you would rent a bike though) or get a taxi to take you there and drop you off. I don't know if a public bus stops there.
Know that it is quite a long walk from that gate to the main palace. Thus my recommendation you think about researching renting bikes. Bikes are a fantastic way of seeing the grounds of the palace. There are places to park the bikes when visiting the Petite Hameau, the Trianons, and the main palace. Riding out through the vast grounds is beautiful and well worth the effort - wheat fields, forest -- lovely.
If you want to walk from the Queen's Gate to the palace, there is a bike rental place just near the Grand Canal before you head up through the gardens to the palace. The bikes can only be taken through the grounds and not into the formal gardens.

We were there last week and decided to do the gardens first also. We had to go back through security for the palace in the afternoon, and it was very crowded!

A “Little Train” runs from the entrance of the gardens to the Trainon and the Queen’s Hamlet. The website of Versailles has a lot of info for preparing a visit. http://bienvenue.chateauversailles.fr/en/gardens/practical-information#!panel-3

I visited the palace two years ago in January, a month with a low attendacy level. First visited the parc, Petit Trianon and the Queen’s Hamlet. After that a few hours before closing went into the Palace with the Hall of Mirrors at the end of the day. With my pre-booked tickets I had only to wait for a family in front of me passing security. I think there were no more as twenty visitors in the Hall of Mirrors, so more “room” to enjoy the place.

Attendacy level in April will certainly be higher, so I think you can visit the Palace as early as possible in the morning and go back to the entrance of the Palace just before closing time to see if the lines are short enough to get in again just to see the Hall of Mirrors. You can always ask ofcourse an employee (entering the Trianon for instance) how long the line for the palace they expect will be at the end of the day.