You shouldn't have any of the issues you mention there with the latest release. However I would say that timeouts may well be server related also. That you will only really know once you have updated. Of course these things use newer technologies than we used over a decade ago though, so that should indeed make a difference. And we also point out issues as you hit them in the latest releases with the support area of the admin CP. For example, if you are not using INNODB (a large contributor to speed issues when not being used), it will tell you in that section.
We don't actually support upgrading from 3.x any longer, but the upgraders are in there if you want to give the upgrade a go. I would suggest on something that is so old, you do a test conversion somewhere first of all to see the results. Content itself should indeed translate over to the new version correctly, along with avarars you mention. Ranks are slightly different in that while they can, it will need some playing to get them to show in the same way. These days we use achievements in the core platform rather than ranks, which is a much much larger system (think instead of 1 point 1 post, you get x points for y achievement. And you choose what those are). You can see more about the achievements item here
We also added more to it here
You can choose whether or not you keep your own ranks when you upgrade, or whether you ditch those in favour of the new default set of ranks, badges, and achievement rules. You can even import those after the event using the xml files directly if you wanted to keep your own, then decide to switch afterwards.
Of course with an upgrade of this length of time, things are quite a bit different in places. I would however very much encourage you to make the jump to our latest products, as I think you will find they benefit you a lot.
If you look at this, get to the point of testing and you think "This isnt something I can do", its worth checking out the providers section. There are quite a few people who offer legacy upgrades as a service.
https://invisioncommunity.com/third-party/providers/
If you have any further questions, please let me know and I will try to answer as best I can for you.