i'm sorry, i'm not a parent, but here's what my parents did for me.
they didn't know it, but i started going on group dates when i was 12. it was me, the boy i liked, and all of our mutual friends (in total like a group of 10) going to a movie. we didn't do anything, except we held hands once. scandalous, right? :) i would let my daughter go on group dates, with lots of people, during the day, at the same age that i did.
not go on single dates at 13. i would save that for 16 or so, but also depending on her maturity and if i liked the guy or not. i started going on single dates when i was 14, but it was just to go get coffee and the coffee shop down the street from my house during the day. but it was just me and my boyfriend.
yes, i would let them have a boyfriend/girlfriend, as long as i liked the kid, and we had already talked about safe sex, relationships, dating safety, etc. (see below).
spend time alone, yes. at our house at night alone? probably not. but i don't see the harm in letting them go get coffee or hang out at our house without me in the room. i started doing that when i was 14.
go to the mall with a friend, of course! same with going to movies with friends. go to the mall and to a movie alone, yes, as long as they had a cell phone. my parents let me go to the mall alone when i was 12, movies alone when i was 13. i also had a cell phone and i'm really mature for my age, and it was always during the day.
r rated movies, yes, depending on what they are rated r for. if it was for like, graphic violence, maybe not, but for language or something, of course. i sort of disagree with the rating system. i saw my first r-rated movie when i was 13 (little miss sunshine).
pg-13, of course. i saw my first pg-13 movie when i was 7.
if by "unrated" you mean nc-17, no, probably not. "unrated" movies mean that they don't have a rating--it doesn't always mean that they're bad. i haven't seen an nc-17 movie.
youtube, i don't really know. i would check what they were uploading and looking at. i would let them have a myspace or a facebook, if they wanted one though. i'd check up on that too.
if my daughter got pregnant at 13, she would be getting an abortion, hands down. if she really opposed it, we could go volunteer together at a shelter home for teen mothers. i volunteered at one once, and it made me not want to have sex until i'm 45 and married. and i think it would change her mind about keeping her baby.
of course i would teach them about safe sex. i've already decided that my children will have a box of condoms in their bathroom, and if i have a daughter, she will go on the pill, because i never want my children to have to go through the anxiety and pain and suffering that i did as an adolescent about pregnancy scares, etc. i want them to be so educated about safe sex, relationships, dating, etc. that they could outsmart a planned parenthood educator. knowledge is power, and as a teenager, i can safely say that sex education does not make kids go out and have sex. it informs them about what to do to protect themselves if they choose to. because it is, after all, their choice, and their body.
when i was 13, my curfew was 10 pm.