No problem to keep your PSE7 and have PSE11 at the same time : you only need to use the organizer in PSE7 to create a full backup, install PSE11 in the new computer and from the PSE11 organizer, do a restore to the new computer.
Since the default 'My Pictures' or 'Pictures' default Windows folders have different structures, it's better to choose the restore option to 'custom' location. For instance you create a master folder ''My new pictures' on your new computer, under which the folders tree will be fully restored (if you check the option to restore the tree structure).
In a first step, the files and folder structure will be restored, in a second step, the catalog will be automatically converted to the new pse11 format.
Very important : Once you have done this, both catalogs are totally independent. What you are doing in one is ignored by the other. You have to think about how you'll have to input and organize new files on both computers.
If that appears to be a big problem, you had better use another option : first install PSE11 on your old computer (you are allowed 2 activations) and PSE7 and PSE11 can happily live together on the same computer. Convert your catalog within PSE11, do a backup and restore on an external drive which can be plugged in alternatively on both computers. The shared PSE11catalog and photos on the external drive has a small drawback : it's slower than on the main drive.
If you want to work mainly on the new computer, the only way to update the catalog on the old one (with PSE11 installed), would be to make a full backup from the new computer and a restore to the old one. You'll have to take care of the difference in folder tree structure and be sure you have enough free drive space; just ask for more details.