Hi Mark,
I hope you will enjoy with our VPS plan. Let me to try to reply on your questions as their understanding may help you even after sign up

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1. I an currently adding some auction sites that use php scripts to query eBay's XML interface. On my LP shared host site they are painfully slow. On a friends VPS site (another web host company) the exact same scripts are instantaneous. My question is should I expect better network performance on VPS?
That's right, you can expect performance improvement for your scripts on VPS as you get more recourses available as well as the hardware node hosts far less sites than shared servers. However, please note that VPS does not have a lot of RAM available so you should be sure that the scripts are not eager for memory.
2. Is there a way to understand the performance problem on the shared host (are there any tools available)?
The first thing to check would be the output of the scripts to find what operations take a lot of time. If the script does not support logging, it can be really hard to find the reason for the slowness without further knowledge what actions it performs.
3. I see that I need to pay extra for Plesk, and that cpanel is not a option on the signup form. Does that mean that cpanel is included and plesk costs extra, or does it mean that I need to pay extra if I want cpanel or plesk?
VPS comes without any control panel. At the stage of the VPS testing, we found cPanel to be not enough stable in VPS environment and thus it was excluded from the plan pricing. cPanel can be added at extra cost, but at your risk.
4. I understand that shared hosting cannot be used for more than 10 sites. Is that right?
One shared account can't have more than 10 addon sites. But you can always purchase another account and upload 11 sites (one primary + 10 addon domains) more.
5. I see that there is a need to register DNS myself. What does this mean if all my domains have already transferred to LP? Does LP DNS then take care of it?
Even if Lunarpages is your domains' registrar, you will need to register your nameservers with VPS IP in order to use them for your domains on VPS server. This procedure is easy and well documented on this forums. We will be happy to assist you with this as well if it will be required.
I hope this helps. Please feel free to ask if you have any more questions.