I don't think a TA or a teacher being strict should be an issue. There are huge differences between different teachers, not just between TAs and teachers, when it comes to how strict a teacher/TA is. So long as she/he is being fair and consistent, a bit of strictness shouldn't be a problem. I am probably more strict than some of the teachers/TAs I work with. However others will also be more strict than me.
I teach ICT for 2 of the classes I do PPA for. I plan all the ICT for y2. I deliver it and assess it for 2 of the 3 classes. In the other Y2 class I teach RE - this has been planned by a different teacher. We share planning between the teaching staff. I also do a half session of RE for one fo the first two classes in my time. In foundation stage I follow rough plans that the class teacher leaves as she finds that most useful because of the way our foundation stage works. They are not full detailed plans - just a list of routines for the afternoon, plus a some numeracy ideas and a focus activity. This is HLTA work in my school. I get PPA time for this.
I also, along with our other TAs, plan for, deliver and assess a phonics group every day. That is TA level 3 work in my school. There is no official planning time allocated for this though. We do have plans to follow but we need to decide what resources, which sounds, etc. to do.
And I plan, deliver and assess an early morning group which is a group focusing on lower middle abilities literacy skills - and both child and parent attends. This happens 3 days a week, and runs for just under a year. I get paid TA level 3 for this, although get no planning time as such for this.