Can I use a 2005 iMac as a target display for a late 2015 iMac?

The 2005 iMacs don’t have Target Display Mode - that did not come along until 2009.

A 2005 iMac would have a 17” or 20” screen with a resolution of less than 1920x1080 pixels. A modern low-end 24” 1920x1080 monitor might go for as little as $70 (more if you wanted one with an IPS panel and near-100% coverage of sRGB). There may not be third-party workarounds for such an old iMac, and even if there are, they would be a waste of money that could be going towards a real monitor.

The Roku stick idea won’t work, either. I suspect the Roku stick plugs into a HDMI port - not a USB one. In any case, the 2005 iMacs had no video input ports.

In a word, no. Please refer to Target Display Mode which has been dead for many years now.

If your 2015 iMac needs an external display, please just buy one. This is:

• Much easier to connect and maintain.
• Does not require difficult to find cabling
• Far more reliable than Target Display Mode ever was.

Yes there are other solutions however they will not save you money. You can search for and find third party apps that allow you to use displays as external displays for other devices. The upside is yes, you can buy these the downside is:

• More difficult to install and maintain than an external display.
• Do not get the same performance as an external display.
• Typically more expensive than an external display.

You can find external displays for less than $100 pretty easily, that is what I would suggest you do. Another suggestion is go to your local thrift store and buy one, they are normally VERY inexpensive, you just need to buy the correct cables if they don't come with them.