recommending a to-do app
Your problems sound familiar: an antique PC at work means that promising new apps such as asana, producteev, zendone or nirvana will not run, or at least not run smoothly. Producteev may actually be the only app that comes close to meeting all your requirements. Doit is quite well-designed, but I find it glacially slow.
If I were you I would give ToodleDo another go. It does require a modest annual subscription but is actually a very robust app. I have used it at work for complex project management tasks for a number of years - see my recent
review
Nirvana is also pretty good and handles the separation of areas (work/home) better than ToodleDo - but it does not always play nicely with IE8.
A modest but effective task manager is Todoist. It is browser-based, has a basic iPhone/iPad version that synchs via the cloud and a low annual subscription fee. You may want to try it out. I find entering tasks a bit laborious though and it doesn't handle recurrent tasks terribly well.
At home I am currently using Firetask. No browser version, but a powerful and visually pleasing app that has made me walk away from omnifocus after a number of years. You can read my recent
review for more info.
Good luck with your quest!
Ozengo