Has anyone removed the spindle before? Will I have to compress the coil spring?

you will not need a spring compressor IF you have a good floor jack, and you are careful, but you will need to pop both ball joints, and a new spindle may not come with a ball joint installed, so you are going to need a new one... you're going to need a ball joint press, and a ball joint separator.

Answer
 

The EPIC has BEGUN... Project "Sway" is Underway!
http://www.peachparts.com/shopforum/bodywork-repair-paint-tools-tips-tricks/138289-w126-epic-has-begun-project-sway-underway.html

Ball Joint Nightmare - chapter 2
http://www.peachparts.com/shopforum/diesel-discussion/173149-ball-joint-nightmare-chapter-2-a.html#post1355788

Advanced Auto Ball Joint Press Tool
http://www.peachparts.com/shopforum/diesel-discussion/267618-advanced-auto-ball-joint-press-tool.html#post2362384

Lower Ball Joint Replacement Question
http://www.peachparts.com/shopforum/diesel-discussion/142212-lower-ball-joint-replacement-question.html#post1065304

Lower ball joints can be done without the MB tool
http://www.peachparts.com/shopforum/diesel-discussion/58908-lower-ball-joints-can-done-without-mb-tool.html#post358030

Write up - Ball Joints & Lower Control Arm Bushings & Tool Review (PICS!)
http://www.peachparts.com/shopforum/diesel-discussion/179284-write-up-ball-joints-lower-control-arm-bushings-tool-review-pics.html#post1418997

Sway Bar issues, front and rear.
http://www.peachparts.com/shopforum/diesel-discussion/282083-sway-bar-issues-front-rear.html

Ball joint broke in half while parking (Pictures)
http://www.peachparts.com/shopforum/tech-help/236457-ball-joint-broke-half-pictures.html#post2004672

300TD suspension rebuild progress
http://www.peachparts.com/shopforum/diesel-discussion/60560-300td-suspension-rebuild-progress.html#post370364

w123 Upper Control Arm R&R- guide joint puller??
http://www.peachparts.com/shopforum/diesel-discussion/146580-w123-upper-control-arm-r-r-guide-joint-puller.html

What's in your balls? upper control arm joint cut open
http://www.peachparts.com/shopforum/diesel-discussion/176291-whats-your-balls-upper-control-arm-joint-cut-open.html

Veteren coil spring compressor users
http://www.peachparts.com/shopforum/diesel-discussion/39182-veteren-coil-spring-compressor-users.html#post222969

group purchase Klann Coil compressor
http://www.peachparts.com/shopforum/diesel-discussion/61677-group-purchase-klann-coil-compressor.html#post378164

Do I really need a ball joint press and a ball joint separator? If so, I'm probably going to have to just let a mechanic do it..

harbor freight separator = $13

BJ press rental from autozone = $0.00000

satisfaction of fixing your own car.... mildly entertaining.

Unless you like blowing a weekend pounding on stuff, I'd leave this to a pro. Suspension work is hard and heavy. And it can be dangerous.

I am going to go out on a limb here. If your Old Ball Joint is good it could be reused. But, only if it comes out of the Old Spingle easy and you do not tear the Boot up.
You can use the Autozone Rental C-Press to Press the Ball Joint in but you need to remove the Ball Joint Boot and it is best if you have a large Washer to put between the Joint and the Press.
See Post #43
http://www.peachparts.com/shopforum/diesel-discussion/280385-lower-ball-joint-1%3B-pos-harbor-freight-vice-0-a.html?highlight=C-press
There is better threads on the above that is the one I found the fastest.
Also in one of the Threads a member has a way of using 2 hammers to pop off the lower Ball Joint from the LCA.

If your ball joints are nice and tight you can replace torn boots rather inexpenxively.