Teaching style should strongly depend on the student, else you risk either wasting time on ideas that are already understood, or on a method that won't get through. Determine what the student does and doesn't already understand, and work from there; guiding a student towards figuring out the correct answer is usually much more valuable than simply spouting information.

Get to know the student, ask about their learning style, and what they think I am going to be able to help them with. When tackling a specific subject for the first time, I would ask the student to demonstrate how they have been going about the subject, to see what can be corrected, and also what solid foundation already exists.

The answer is twofold: by helping the student see the topic as interesting/not-intimidating, and by not simply giving away answers when helping students solve problems. A chore is tiresome, but a puzzle is rewarding, and if students can see a problem as a game they are capable of winning, that can be a powerful motivator, and they will happily accept your help in developing the tools to do so.

This will completely depend on the student, but the general answer is by finding the source of any discouragement that might arise, helping to remove it, and finding the source of the existing motivation and helping to encourage it.

This is another answer that will strongly vary student to student. For some, more examples are all that is needed, while for others, a complete change of strategy might be what works best. While usually it is best to guide students towards figuring out problems themselves, when someone is stuck, it can be helpful for them to see a demonstration.

It depends on if the goal is to help the student understand the given passage, or to improve their comprehension overall. Still, some components remain the same - find out if any vocabulary is unknown, and explain those words. Then, help the student look at context to figure out meaning. Not understanding text can be particularly discouraging, so it is important to be deliberate during this process.

It is always best to start with finding out who the student is and what they know, rather than to have a rigid structure already in mind. They already have a teacher who is giving them inflexible lectures. If they need additional help, that should be tailored to whatever gaps still remain.

Demonstrate examples of what comprehension of the subject can mean. In addition, by trying to guide the student to solve problems themselves, the almost universal drive to see a puzzle through to the end can come into play.

In addition to having them actually solve example problems, I find having the student explain the material to me once they think they understand helps both to ensure that they actually understand, and actually aids their retention of the material - people remember what they have said better than what they have heard.

Carefully monitor difficulty such that they encounter more problems that they can do than ones that they can't. However, the tone of interaction is just as important. Even if the student is frustrated, the tutor can never be.

I find that asking them is a good first step. Combined with actually seeing how they perform when presented with the material, this can give me a good sense of what is required.

I don't so much adapt to student needs so much as build my tutoring based on student needs. Changing to suit the student's needs isn't step 2 or 3 - it is step 1.

It depends on how the student learns. However, pencil and paper is always useful, since people remember things better when they write them. Plus, it can help me diagram things out to help visual learners.

Having Google present can be helpful, because demonstrating how to search for information one doesn't have goes a long way towards helping students become independent learners.