Can a current earth-made spacecraft alter speed and / or direction in space without using another object to do so?
Q: Can a current earth-made spacecraft alter speed and / or direction in space without using another object to do so?
Q: From the little I understand, our present era spacecraft have to push against another body - such as a moon or planet - to alter speed and direction.
No. They do not. Your understanding is in error. Although pushing against another body can be very helpful under some circumstances, it is not required and is very much the exception, not the rule.
Q: Could one of our spacecraft slow down, stop, start moving from a standstill, alter direction or speed up if in deep space with nothing around it?
Yes, though stopping and reversing direction is something we keep to a minimum because it takes a lot of energy (and I should point out that "standstill" is always relative to some other object.
You seem to have a good grasp of Newton's laws, but a fundamental flaw in your understanding of rockets. Yes, rockets absolutely need something to push against, but that something isn't the air or the Earth and another object, it's their own propellant. With very few exceptions (some of which I'll address below) all rockets used today (in space and on Earth) operate principally by ejecting a small mass at high velocity (rocket exhaust) in order to accelerate a larger mass (the ship) more slowly in the opposite direction. The momentum (mass times acceleration) of the propellant always matches the momentum imparted to the ship.
Consider the Saturn V that carried men to the moon.