1. That depends on the degree of the condition, and on the intensity of your work. You can execute any Thai Massage technique in a very strong or very gentle way. Yes, you can work on the shoulder and the arm pit if there is bursitis if you match the intensity of your work with the ability of the client to receive it. 

You need to get the client to guide you. The best way to do that is to use the one-to-ten method where you ask the client how a particular technique feels on a scale from one to ten. You want to work in such a way that there is some discomfort that feels releasing, but no real pain. This tends to be in the 7-8 range on the 1-10 scale for somewhat normal conditions. For more severe conditions you might have to work lower on that scale.

However in the case of bursitis I would use motion based techniques like circling or rocking or back and forth movements instead of linear pressure techniques. The concept here is that you don't want to do some massive stretching, but you want to warm, loosen and relax the shoulder and you want to re-establish a feeling of well being. You are after changing the cell memory of the affected area rather than trying to change a physical condition on a physical level. Your goal with the bursitis is to free up blockages, re-establish energy flow and change the cell memory from pain to well-being, even if only temporarily.

2. You can develop a high degree of feeling in all body parts which you can use for Thai Massage. However there is no doubt that you have the most sensation in your hands. That's why it is a good idea to probe for problems with the hands, and then use other body parts to treat the area.

Just to clarify - you do not press, you lean with body weight. Pressing generally causes discomfort or pain, whereas leaning feels good.

Done correctly, clients should hardly be able to tell which body part you are using. No matter which body part you use, it should be a pleasant sensation for the client, even if it is a somewhat intense technique.

3. Yes you can work on the hamstrings in the case of sciatica. However you have to remember that the cause or the origin of the sciatic condition is not in the legs. That's where pain is often felt because the sciatic nerve runs down the back of the legs.

There are two areas where sciatic pain can be caused or start out from. One is the last couple of lumbar vertebrae where the sciatic nerve leaves the spinal canal. That's where it can be compressed by a variety of conditions.

The second area is what's called the "piriformis syndrome". This is where the sciatic nerve is being compressed by the piriformis muscle. 

So working on the hamstrings can help and feel good, but it does not address the sciatic pain at the root of the problem. This is a big subject. I have an entire 4 hour course just about treating sciatica with Thai Massage. So I can only touch briefly on the subject here.