Several problems with 1986.5 Mk3

[essial]

I just bought this car from my step brother (because I have the money to fix it up and he wanted a car he could just drive and not worry about) so I've come up with a list of issues and fixes. This supra has been modified a bit, namely, I've been told the transmission (it's an automatic) has been replaced(?) with a sports transmission (when you put it into drive or reverse it kicks it hard into gear, and shift like that too).

My first question is about engine power. It seems that after about 3,800rpm, the car gets a nice kick in the butt and really takes off, but getting up to 3,800rpm feels like driving a minivan (it's like two totally different cars below and above that mark). A second (and possibly related) issue is that for the first minute or two after starting, the transmission refuses to shift out of first gear. After the first few minutes it shifts great. I've been told that a new throttle position sensor has been installed (before, it simply would not shift out of first at all, ever), and set dead center (which explains why it idles at around 1,200 rpm). I'm going to adjust the sensor to the proper settings this weekend (gotta start learning somewhere, and I got the manual), but was wondering if the shifting and weird power issues are related to that, or something else.

Secondly (and this may be very simple). Again, the first few minutes after turning it on, it has this horrible screechy sound like you hear with some clunkers going down the road. Pressing the gas (in park) will make the rate of screeching increase. After a few minutes, sound goes away and never returns (though early on sometimes turning makes it start back up for a few seconds). I do know the steering pump hose is leaking and thus there is no steering fluid (and thus no power steering) in the car right now, which I plan on replacing once I snag a new pump hose (the shop wanted $300 to replace the hose, no thanks). Just wanted to make sure that this is the most likely cause of the screech and not something more serious.

Anyway other than a few startup problems, the car is great fun to drive, and it feels great being one of the only stylish 80-s style cars I see out on the road. It's already been lowered and has low profile rims (yeah, teenager previously had it), so I'm mainly using it as a spare car, and as a way to learn how to work on cars. Although it's a bit annoying with the extra attention it gets.