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Adding a Non-Model Form in Django Admin

I am trying to add a non-model form in django admin interface and am not able to find any particular way to do it. This form would do some processing and change some data in the DB. But this is not related to a particular Model and should stand out. This form should not be available for the user to use.
One thing I can do is add the form to the general view and prohibit using permissions but I was thinking since django admin interface already exists, it would be better to add that to the django admin interface.
Is this possible to do in Django?

You can add arbitrary views that within a ModelAdmin that do whatever you want. See the documentation for ModelAdmin.get_urls. You can do the same at a higher level by defining AdminSite.get_urls.

Conditional Logic in django forms

I'm trying to build a conditional form where selecting one option will cause a new field to appear underneath, while selecting another option will display something else.
Formstack has a good example https://www.formstack.com/features/conditional-logic
I couldn't find any preexisting form packages for django with this functionality. How should I start implementing this?

Django forms (especially if you use the ModelForm library) are a direct reflection of your Django application Model. You should therefore start by refactoring your Django application Model to have fields that have optional values (i.e. they can be NULL, empty or have a default value already created).
These would be the form fields that are shown/hidden based on your conditional(s) and they may or may not have values (if they are hidden based on a conditional it is impossible to provide a value to them so the Model fields must be able to accept NULL values or use the defaults).
You would then use a client-side language such as Javascript (JS) to handle the user iteraction with your Django application. A simple to use JS framework like jQuery would be worthwhile investigating for your needs.
In addition to the exceptional Django docs on Forms, I also highly recommend you take a look at Django Crispy Forms writtten by PyDanny to see how Django forms should be done right.

When to use django forms vs manual forms?

I am beginning learning django and have just come across django forms (I've always used manual forms in the past in django templates).
Should I always use django forms no matter what or are there scenarios where I should write the forms in the templates manually?

You can use your own forms, but it is better to use django forms because it provides more customization and flexibility.
It is easy to change the type of "widget" rendered with very little code change when you use django forms.
For example:
class MyForm(forms.Form):
my_field = forms.ChoiceField(choices=CHOICES) #choices can be a tuple
Would render as a checkbox field.
If you want to render the same as a Radio button, all you would have to do is:
class MyForm(forms.Form):
my_field = forms.ChoiceField(widget=forms.RadioSelect, choices=CHOICES)
You would have to manually change the entire code if you have your own form.
You can also do some custom field level validation on the server side in one pass (Example, check if username is unique) - You can achieve the same in a custom form, but you will have to handle every scenario yourself.
Also, django has ModelForms which would be a replica of the Model object - that tremendously reduces the amount of work that needs to be done for validation, and form processing.

Improving Performance of Django ForeignKey Fields in Admin

By default, Django's admin renders ForeignKey fields in admin as a select field, listing every record in the foreign table as an option. In one admin-accessible model, I'm referencing the User model as a ForeignKey, and since I have thousands of users Django is populating the select with thousands of options. This is causing the admin page to load incredibly slowly, and the select is not very useful since it can take a while to scroll through thousands of options to find the one you want.
What's the best way to change the rendering of this field in order to improve page load and usability? I'd like the select field to be replaced with some sort of button to launch a search form popup, or a text field that searches keywords via Ajax to find the Id for the specific User they want to associate. Does admin have anything like this builtin, or would I have to write this from scratch?

Add raw_id_fields to your model to only show the ID instead of a dropdown.

You're right, Cerin, the cause of the slowdown is because Django is populating the <select> element with too many options. You might want to use an autocomplete element instead.
Interestingly, Django 2.0 has introduced a new feature on the admin site, called autocomplete_fields, which I think you will find useful in this case. It uses AJAX.
class ExampleAdmin(models.ModelAdmin):
autocomplete_fields = ['example_field_user']

You can use one of the few autocomplete apps for Django. Check them at Django Packages.
There's also django-extensions that have ForeignKeyAutocompleteAdmin that fit your needs pretty well.

Another option is to add readonly_fields instead of raw_id_fields

Django best practice for displaying mostly read-only form, one field writeable

I have a requirement where one user creates an 'instance' of an object via a ModelForm. Another user of a different group has access to read all of the fields of the form, but has to update only one field. Think of a student who creates an exam object. Then a teach pulls up the exam and just needs to put in a grade, the rest of the exam is read only.
What's the best way to do that? Should I just query for the object, and display each field individually, then create a form (not a ModelForm?) for just the one field?

Should I just query for the object, and display each field individually, then create a form (not a ModelForm?) for just the one field?
This is probably the best way to go about it. Note you can use a ModelForm for the teacher form, see the Django documentation on using a subset of fields on a model form. You will have to display all the other fields manually in your template, but you should probably have a separate template for this view (I would use separate views as well).
You could find some code for a read only field on Django Snippets, but generally it's better to be explicit about what fields you are updating from each view. This is likely to be more trouble than it's worth.