"The theory holds that gravity is geometry: particles are deflected ... not because they feel a force ... but because space and time around the object are curved." Can anyone explain to me why it can't simply be a force? If you accept that gravity is caused by energy and not mass (which is obviously the case), then I see no reason a photon can not experience a force. So what is the reason to insist it's geometry? And…
It could be a force - if someone finds a theory that describes all the observational data. They haven't.
The reason to insist geometry is that general relativity manages to predict extremely accurately the result of all gravitational experiments/observations as of 2015. In science, prediction comes before explanation, meaning if you have to accept whatever explanation the best predictive theory offers and not vice versa.
Well, its has been explained - you just haven't gone through the right books. There are two answers, which I will simplify. One, the Big Bang threw everything all over the place and now masses are moving so as to reach the state of minimum energy. Two, there are 3 other forces in the universe which can cause masses to move. For example you can use electromagnetic forces during the combustion of rocket fuel to send a satellite in space. The satellite then moves in orbit because of the geometry of spacetime. Same when stars explode - this involves all three forces.