I am having issues upgrading to the latest version. I started with version 2.2 and tried to upgrade directly to the 3.2.7 but it did not work, there were messages about multiple missing columns when trying to start the site.

I was able to upgrade to 2.3.1.15 without issues.

I tried upgrading to different point releases of v3, the lowest being 3.0.2 but even that gives the error message when starting the site:

Exception type: SqlException

    Exception message: Invalid column name 'PageSize'.

I seem to have an issue with the membership migration going from v2 to v3? 

Also, the main reason (other than old age) for updating is that emails are no longer being sent from the forum even though nothing has changed. This email issue also persist s with 2.3.1.15 version. There is an error in the log "System.Net.Mail.SmtpException: The operation has timed out.". I am able to send mail from other services using the same smtp server settings.

I tested upgrading from 2.3.1.15 to 3.2.7. and it worked without issues. 

You wrote that you initially tried to upgrade from 2.2 to 3.2.7 and then to version 2.3.1.15 did you every time restored the db, before upgrading?  If you try to upgrade to different version its possible the the upgrade migration was not working correctly.

However, now I went to the "forum/install"-page manually and was able to run the upgrade process. The database looks to have updated, but now I have a null value error instead when navigating to the forum.

​Exception information: 
    Exception type: ArgumentNullException 
    Exception message: Value cannot be null.
Parameter name: value
   at System.Runtime.Caching.MemoryCacheEntry..ctor(String key, Object value, DateTimeOffset absExp, TimeSpan slidingExp, CacheItemPriority priority, Collection`1 dependencies, CacheEntryRemovedCallback removedCallback, MemoryCache cache)
   at System.Runtime.Caching.MemoryCache.Set(String key, Object value, CacheItemPolicy policy, String regionName)
   at System.Runtime.Caching.MemoryCache.set_Item(String key, Object value)
   at YAF.Core.Configuration.Pattern.TypeFactoryInstanceApplicationScope`1.Get()
   at YAF.Core.Services.LinkBuilder.GetLink(ForumPages page, Boolean fullUrl)
   at YAF.Core.Services.LinkBuilder.Redirect(ForumPages page)
   at YAF.Core.Services.Permissions.HandleRequest(ViewPermissions permission)
   at YAF.Modules.PageSecurityForumModule.PagePreLoad_HandleEvent(Object sender, EventConverterArgs`1 e)
   at System.EventHandler`1.Invoke(Object sender, TEventArgs e)
   at System.Collections.Generic.List`1.ForEach(Action`1 action)
   at System.Web.UI.Control.OnInit(EventArgs e)
   at System.Web.UI.Control.InitRecursive(Control namingContainer)
   at System.Web.UI.Control.AddedControl(Control control, Int32 index)
   at YAF.Web.Controls.Forum.OnLoad(EventArgs e)
   at System.Web.UI.Control.LoadRecursive()
   at System.Web.UI.Control.LoadRecursive()
   at System.Web.UI.Control.LoadRecursive()
   at System.Web.UI.Control.LoadRecursive()
   at System.Web.UI.Page.ProcessRequestMain(Boolean includeStagesBeforeAsyncPoint, Boolean includeStagesAfterAsyncPoint)

I also noticed that I am limited to 3.1.15 because I am on Windows Server 2019 which does not support .net 4.8.1 (unless I go v4, but that is not fully released yet).

Okay, that’s understandable. But there where some isssues in the first releases of 3.x regarding updating.

But I don’t think upgrading to the latest version will resolve the mail issue for you.

does it not send any emails? If so then have you enabled ssl for the mail settings?