Assuming that your supervisor proposed this research idea and you agreed to do the research for them.
(1)You wrote 80-90% of the paper and performed the experiments.
Of course you discussed with the supervisor as you went along, but you conducted the experiment and wrote the paper on your own.

Who should be the first author of the paper?

(2)After that paper, you got this idea to extend the initial idea; you conducted another experiment and wrote another paper on your own.

Who should be the first author of the paper?

This is similar to what I have recently done. The first paper, due to be submitted soon, has me as first author and my supervisor as second author. There is also a 3rd author on mine who is the person who created the computer model I used. he had no real input but as he is one of the 'big names' in my field my supervisor said it would help my paper to get noticed. I wrote the entire paper, ran all the experiments and did all the analysis with guidance from my supervisor, he suggested some ideas and a few changes to the wording of the paper.

With regards to the second paper you should be first author with sup as 2nd author if appropriate. I don't know what field you are in but I am in science and this is how it would work for me as it is rare to be sole author on a scientific paper.

Generally the person who puts in the work and effort should be the first author. The supervisor can be made the corresponding author (which can be placed at the end if there were other contributors to the work) as he proposed the idea. Both these positions are considered equally important.

Ah, I have another constraint; what if the project is workable only if you use the supervisor's grant or use his facilities?

Generally a position for PhD researcher is filled by a candidate after being selected by the supervisor based on his resume and abilities. It is usually the case that the candidate will work in the lab and use the facilities of the supervisor and the department. With this in mind, my previous answer holds true.

The content of the paper evolved from initial experiments that I did at the very beginning but much of the content was my ideas with advice and it was sup's idea to write it up as a paper after seeing the results.

I still think that if you have done the experiments, analysis and written the paper then you should be first author, after all it is your work. The supervisor helped that work to happen so should probably get named as second author in order to credit their contribution

This is a VERY tricky question my friends, and the answer hinges on what can be variously read as:

Intellectual propriety and due defference to intellectual leaders or

Territorial pissings and which baboon has the biggest reddest ass.

Tricky question, depending on the "need" of your superviser to get papers published for his own career. Rules in our lab are: If you do most of the experiments in the lab and contribute massively while writing the whole thing up, you should be first author. Supervisors normally get the corresponding author, but this can be also very problematic if you have more than one supervisor which all "need" a paper for their annual output.

The lab thing is irrelevant in my eyes. Its the job of a supervisor to offer space and materials for you to work. Especially with grants, they get paid to teach and train new students, so what should a PHD student in science do without a lab? Setting one up at home? If your supervisor is too eager to get the first authorship, he or she should think about the amount of labwork, you have done. Sureley, he/she could walk into the lab and do things him/herself, but then he/she hasn't the time to write applications for new projects, supervise other students and get more papers published.

So in my eyes its fair for you to get the first position on both papers (even if it wasn't "YOUR" idead), and your supervisor should get the corresponding author for both because of supervising you and offering the ressources for your research.

The only exception I would make is when your supervisor needs one paper badly for his/her habilitation - then I would step down.

Oh and btw, the question gets even trickier, when Technicians are involved that have done a considerable amount of work for the paper :-D