I'm going to agree with the others, in that your server is likely overloaded. The main goal with shared hosting is maximizing the site/server count to increase profit. The more sites you put on a server the slower that server is going to run, as there are less available resources. In addition if you have bad neighbors (other sites) things are going to run even worse until they are addressed by the provider.
Ultimately, there is nothing you can do to correct problems with a shared server as you will lack the necessary access. You can request they move you to a different server which might perform better, however it's typically a matter of time before that server will exhibit problems. I've noticed that providers making use of custom control panels (ex: GoDaddy, 1&1 and Powweb among others) tend to have these problems more frequently.
That being said if you have SSH access, you can run some basic commands to gauge if that server is overloaded. The three commands you will need to run when the issue is occurring are "free -m" "cat /proc/cpuinfo | grep processor" and "top" (without quotes). They will provide you with the current memory usage, number of processors, and the server load at that point in time.
In addition with 1&1 the database connection will need to be factored into the problem, as the database is separate from the web server which introduces additional delays. Any request interacting with the database will be subject to their current overall network status, as your request must go to another system, be processed, and the data returned. This is less than ideal as there are more things that can go wrong making troubleshooting the problem more difficult.
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Now that being said I do want to touch on your statement regarding MODX and speciality hosting. Speciality hosting is NOT required with MODX, however resource stingy hosts will not typically cut it. You do tend to see higher resource consumption with MODX, but nothing extreme by any stretch.
Yes, MODX Cloud is a speciality hosting option - however it's purpose built for MODX Development. If you want a good list of MODX Friendly hosting providers check out
http://bobsguides.com/modx-friendly-hosts.html which lists several great providers. You may pay a little more with some of these providers, however the hosting will be of a higher quality with better support in my opinion.
I'm not saying that MODX can't run on 1&1, however you might have a far easier time using a different provider (and closer servers).