It's not docman specific. You just have to install SSL support for Apache (or your web server) and it will work (take a look at http://www.modssl.org/ for more info about installing Apache with SSL support)
DocMan uses standard Apache icons (which are under /icons/ URL if you have Apache) or symbol font (available only under windows I'm afraid).
php3 isn't supported (intentionally). You should really upgrade to php4. From version around 1.8 there are so many dependencies on php4 that you don't even want to try to port it back to php3.
php has two limits on uploading files configured in php.ini
upload_max_filesize is set to 2Mb by default, you should
increase it. post_max_size is set to 8Mb by default and
should be also increased for large files.
memory_limit is 8Mb and you should increase that one too, and max_execution_time is by default 30 seconds. If your connection with docman site is slow, increase that also.
If previous answer didn't help and you have RedHat (and possibly other distributions) changing LimitRequestBody in /etc/httpd/conf.d/php.conf will help.
It's a design decision to overcome some of problems with http authorization. You will have to login as different user to continue working.
Version 2.0 of docman should work with php 4.2.0+ without registering of global variables (register_globals off option in php.ini).
Yes, in primitive form. In configuration file you have option $gblNumBackups (by default it's 3). This means that docman will keep 3 last versions of documents which are updated via check-out/check-in procedure. You can retrieve old versions by clicking on properties (icon left of filename) and then selecting version from "Changes to this file".
See also: Installation instructions, Upgrade from v1.x, New Directory Layout and Administration manual