IPhoto

Yes, larger libraries will take longer to open. But no Library will take long. I have more than 50k items in mine and it opens in a moment.

Define 'Archive' in this context.

You can't run an iPhoto Library in the Cloud.

You can run a library from an external HD as long as it is a wired connection and the disk is formatted Mac OS Extended (Journaled). It will make a negligible impact on running the library.

Does the volume of photos in IPHOTO affect the time it takes to open the app?

iPhoto can handle 1000000 photos. iPhoto is a bit slow on opening, when you close iPhoto with a product selected - a slideshow or a book with many pages. Then it will try to open the product when it launches again and waste time. Or if you are syncing with Photo Stream, iPhoto will try to download the new photos from the Photo Stream when it launches. But the size of the library will not matter much.

Secondly, if I archive photos into the cloud or a stand alone drive, would that impact the performance of IPHOTO?

Don't store your iPhoto library on a network volume or in the cloud. That is not supported by iPhoto or Aperture. This may corrupt the iPhoto library and you will risk to lose your photos. (See: Use locally mounted Mac OS X Extended volumes for your Aperture library)

Your iPhoto library should be on your system drive or a locally connected drive. Use a drive with a fast connection- USB 3 or Thunderbolt. If the port is not so fast, you may notice a slower performance, but not dramatically slower, like you would experience on a network volume.

I shall exercise patience.

What is the problem?

Is iPhoto slow on your Mac?