I'm not sure about all your requirements, but I would recomment to look deeper into ZOC SSH Client. I have used it in the past (still occasionally) and I can confirm a few of your requirements:

• running on 7/8/10 (per their feature page)
• you can choose a custom data folder on their installer
• has an extensive directory of hosts with stored password or scripts (the directory is a single file, I often copied it between laptop and PC via File > Show Data Folder)
• generally superb scripting support
• logfiles for sessions (also debug logs)
• commandline parameters for nearly every thing
• can also create quick launch icons on the desktop from host directory

I'm not sure about all your requirements, but I would recommend to look deeper into ZOC SSH Client. I have used it in the past (still occasionally) and I can confirm a few of your requirements:

• running on 7/8/10 (per their feature page)
• you can choose a custom data folder on their installer
• has an extensive directory of hosts with stored password or scripts (the directory is a single file, I often copied it between laptop and PC via File > Show Data Folder)
• generally superb scripting support
• logfiles for sessions (also debug logs)
• command-line parameters for nearly every thing
• can also create quick launch icons on the desktop from host directory