Hi Austino
You have walked right in to one of the major fun parts of owning a woodland! That everyone has a different opinion.... if I was you, I would call NE, or the local FC officer and talk it through with them, get it straight from them - not the policy that you read.
In terms of a work shed and tree house - all woods have restrictions on what can and cannot be built without permission. It might not be any different in a SSSI.
You can have a tool shed/ store for forestry purposes when you can show it is necessary. In other words - you won't be able to build something that contains a bed and a kettle - but if you can show that you need shelter to store tools for forestry purposes then you will be fine. These are the kinds of things you can work through pretty easily once you own the woodland.
If you love it, I would call the NE folk and talk to them and then go for it!
You won't have a problem selling it. I don't think 50 % of people wouldn't touch it - this forum is a very very small selection of our members. Many swog members own SSSI and so far I have not had anyone telling me they wish they hadn't bought one with the designation.
It may have more restrictions and more paperwork - but it just might be worth it if the wood is right for you. I know I am not giving you a straightforward answer - that is just because I don't know! Call NE...I am sure with this careful thinking you will make the right choice
Best wishes
Tracy