Just got this one, having some trouble with it.

OS: Windows 7

1; The mouse cursor, whenever I try to move it slowly, it doesn't move at all or it skips.

2; The taskbar is somehow overlapping on top of the game, whenever I click on the bottom half of the screen it switches to the desktop. There was a fix mentioned but it seemed to make the problem worse; I had to put the computer in sleep mode, wake it up then use the log off option to force close the game.

If anyone can offer advice on how to fix these problems, I would greatly appreciate it.

If you want to play the classic FS1 release and only the classic one I can't help you. However, playing Freespace 1 in the Freespace Open Engine (using FSport) is guaranteed to fix your problem. Volition released the source code for Freespace 2 and since then a group of modders and programmers have been working on the engine, improving it, and porting it to modern OSs. It makes the game look a lot nicer and improves compatibility of the game with modern operating systems. It does however require a lot of hard drive space and download time. The automated installer is here: http://scp.indiegames.us/bnr_installer.php. Instructions are here: http://www.hard-light.net/wiki/index.php/Getting_started

Good hunting!

Besides the Freespace Open suggestion (which is still you best bet) if you wanna try anyway to fix the original version, have a look at
http://www.gog.com/en/forum/freespace_series/solution_afterburner_briefing_screen_stall_input_lag

This may be connected to a problem where a second mouse pointer appears below and to the right of the game mouse pointer. The second pointer appears to be in the windows desktop. When you click the game mouse pointer, the pointer in the desktop also clicks, activates the desktop and stalls out the game (and locks up the computer).
I found that if I start the game by going into the game folder wherer the files are and opening Freespace.exe, --- the game console appears. If you click on settings and redetect the video card (I set it to fast clear for cut scenes) and joystick (I make sure the force feedback options are unchecked), and start the game from the console (Click the Play FreeSpace button), the game starts, the second mouse disappears, and you can play the game with no problem.

I can confirm that this was what my game's symptoms were and fixed this.
In fact, all I did was 1) open Freespace.exe and 2) disable force feedback on the joystick tab.
It runs perfectly now even when I start the game the normal way.

Win 7 x64
AMD R9 290 (Catalyst Version 14.12, Driver version 14.501.1003)
Dual 24" full HD screens
No joystick; using keyboard and mouse.