Hi, Pro Boxing Dad. I can add a little to Alonya's answer. She speaks for MLO, I'm just a user like you.
It's true that you can't inherit categories from a view filter. In the simple case you described it would work, but consider this case: I have a context @CallsHome and @CallsWork. I have a view that filters on Context contains '@Calls'. If I created a task while in that view, MLO would have no idea whether to use @CallsWork versus @CallsHome. Other filters are worse than category: suppose you had a view showing tasks created yesterday and while viewing it, you created a new task. Would the new task show today or yesterday as the creation date?
Maybe there is a different way. If you don't have any errands embedded in projects, you could consider moving your errands to an "Errands" folder. The folder would have the @errands context assigned. The Errands view would be zoomed in (Ctrl+R) to the Errands folder. If you create a new task while in that view, it will be placed into the Errands folder and will inherit the @errands context from the folder.
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Re your second point, I don't know anything about IOS but I can tell you how it works on Android, often it's similar. On Android, if you click the funnel icon, it brings up a screen where you can filter on a text string and specify if it will apply to title, notes and/or contexts.
For your third point, if you are using cloud sync you can set up something called tasks by email (TBE) where you get assigned an email address at
mylifeorganized.net, and any email you send there shows up as a task, with the task title from the email subject, and the task note from the email body. Once you have that, all you need is a way to send an email out of your browser carrying web page title and URL. I don't know chrome but I bet there are a lot of available tools to do that. On Firefox I wrote my own, I cheated a shortcut linked to the following Javascript:
javascript:location.href='mailto:?SUBJECT='+document.title+'%20&BODY='+escape(location.href);
This script requires you to type in a destination email address (the TBE address) but you could easily add that to the script.
-Dwight