Respawn doesn't work in SP, but it can be useful to quickly test, even an MP-only mission, in SP mode (all MP missions work in SP, or at least they should do, but anything but a very simple SP-only mission is unlikely to work in MP).
Well, a lot of your problems are in fact limitations with the helo scripts you are using, rather than things you should be expecting to just fix up yourself. Get onto the people that wrote them and ask them to make them respawn-compatible! Something else that you might not have noticed, although it isn't a massive problem, is that if you stand near one of the other group leaders, then you will actually have two "helo pickup" actions available. Sadly, you can't assume that scripts you download are perfect and you must check all documentation for known bugs and accepted limitations. Most of the time, however, you will be able to live with the small imperfections of a script, especially if you couldn't write it yourself.
To ensure that the AI respawn, it would be better if they weren't players and you manually re-spawned them. If they could be players, then someone will end up playing as them and mess everything up! Before we start discussing how this might be scripted, you should also consider whether you want the AI lackeys to just respawn next to the player. What if the player is in a fire-fight and the AI just keeps respawning in the line of fire and keeps dying? Anyway, something to think about.
The reason that the "helo pickup" actions disappear is that all actions are placed on an object, such as the player object or a Humvee. When you die, the action stays on your corpse and you don't automatically get the action put on your re-spawned self. This effect has nothing at all to do with player names or group names being "forgotten", but is just an effect of this script you are using not having been made respawn-compatible. If the script was not actually intended to be respawn-compatible, then that is fine, of course.
As to your last comment about MP scripting, the essential problem is that in MP games you have to manage the actions of and synchronise the data on 10 (or whatever) computers all at the same time, whereas in SP, you only ever have one machine to deal with. Although it is possible to make simple MP missions without dealing with a lot of these issues, once an MP mission is non-trivial, it is very easy to get things very wrong (I know, I've messed up countless times already in MP scripting!). For example, in the helo script you are using, the helocopters are managed by the dedicated server, but each player's soldier is managed by their local machine, so, before you can get anywhere, you need to get the computers to start talking to each other...