Explanation
Marketing Cloud Account Engagement can sync with Salesforce custom fields that are on the Lead, Contact, or Account objects. Custom Prospect fields in Marketing Cloud Account Engagement can be mapped to custom Lead or Contact fields in Salesforce. Custom Account fields in Marketing Cloud Account Engagement can be mapped to custom Account fields in Salesforce. Custom Opportunity fields in Marketing Cloud Account Engagement can be mapped to custom Opportunity fields in Salesforce. Custom Record Type fields are not supported by Marketing Cloud Account Engagement and cannot be synced

According to the Salesforce documentation, the place on a prospect record where the user would look to see if a prospect had registered for a webinar is: C) Prospect Activities. A prospect record is a feature that shows the detailed information and activity history of a prospect in Marketing Cloud Account Engagement. A prospect record can be accessed from the Prospects tab in Marketing Cloud Account Engagement, and it can show different sections of information for the prospect, such as details, insights, activities, or custom fields. The Prospect Activities section shows the chronological list of actions and events that the prospect has performed or received, such as opening an email, clicking a link, submitting a form, or registering for a webinar. The user can use the Prospect Activities section to see if a prospect had registered for a webinar, and to see other details of the webinar registration, such as the date, time, or status. The user can also filter, search, or export the Prospect Activities section. The Contents, Insights, or Custom Fields sections are not the places on a prospect record where the user would look to see if a prospect had registered for a webinar, as they are related to other aspects of the prospect information, such as content preferences, engagement metrics, or custom data. Reference: Salesforce documentation

The automation tool that should be used to increase a prospect's score an additional five points every time a specific form is completed is completion action. A completion action is an automated task that is triggered by a certain element in Marketing Cloud Account Engagement, such as a form, a form handler, a custom redirect, a page action, or a file download17. A completion action can perform various actions, such as adjusting score, adding to list, assigning to user, sending autoresponder email, and so on18. A completion action can be added to a form in the fourth step of the form builder tool, and it can be customized to execute only on prospects who meet specific criteria19. A completion action is not retroactive and will only apply to prospects who complete the chosen action moving forward

The recommended way to identify the prospects who purchased a certain product within the last year and export them to a .csv file is to create a segmentation rule based on the Product Name field. A segmentation rule is a one-time rule that matches prospects based on certain criteria and performs certain actions. You can create a segmentation rule that matches prospects who have the specific product name in the Product Name field and the action to export them to a .csv file. This way, you can get a one-time export of all the prospects who have the product name before it changes. An automation rule, a completion action, or a dynamic list are not suitable for this goal, as they are either recurring, real-time, or based on an activity, not a one-time export based on a field value. Reference [Segmentation Rules Overview]

Explanation
According to the Salesforce documentation, a prospect's score can be changed through automation rules, scoring model changes, or completion actions. A prospect's score is a numerical value that indicates the level of interest and engagement of a prospect, based on their activities, such as opening an email, clicking a link, or submitting a form. A prospect's score can be changed by using different methods, such as:
* Automation rules: An automation rule is an automation tool that can be used to update a prospect's score when the prospect meets certain criteria, such as prospect field values, activities, or scores. For example, an automation rule can be created to increase a prospect's score by 10 points when the prospect visits a specific web page.
* Scoring model changes: A scoring model is a feature that allows users to customize the scoring rules for different types of activities, such as email opens, form submissions, or custom redirects. A scoring model can be changed by editing the default scoring model in Marketing Cloud Account Engagement, or by creating a custom scoring model for a specific group of prospects. For example, a scoring model can be changed to assign more points to an email open, or to create a different scoring model for prospects from a certain industry.
* Completion actions: A completion action is an automation tool that can be used to adjust a prospect's score after the prospect successfully completes a marketing element, such as submitting a form, clicking a link, or visiting a web page. A completion action can be used to increase or decrease a prospect's score by a specific amount, or to reset a prospect's score to zero. For example, a completion action can be created to decrease a prospect's score by 5 points when the prospect unsubscribes from an email.
References: Salesforce documentation