I've installed version 3.0.5 to a new folder on my off-line pc. Created a standard wallet_1 (same name as the one on former installation) and restored it from seed. Then, I've overwritten C:\Users\E8-950\Desktop\electrum_data\wallets\wallet_1 to the same path of the new 3.0.5
I'm not sure why you created a new wallet, restored your wallet from seed... and then overwrote that file with your old one? 
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As regards the on-line pc, I've upgraded to version 3.0.5 as well and created the watch-only installation restoring from PMK. Last, I've overwritten wallet_1 to proper path.
And you've done the same thing here? Created a new wallet... then copied over the old one? Quote
When I open my new Electrum 3.0.5 installation I can see my history transaction although they are marked 'Not Verified'
On which computer is the wallet showing "not verified"? Is that on your "offline PC"? because that status usually just means that your wallet hasn't "synced up", so it does not know if the transactions have been confirmed or not. Electrum stores your transactions in the wallet file, so if you copied the old wallet file, it would have those transactions, but wouldn't be able to confirm if they were OK or not.If it is on the "online" PC, then it would seem that your wallet is connected properly? Is the dot in the bottom right corner red, green or is it blue arrows?
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The last question (not necessarily correlated to the above but of some interest to me): Having to do the Electrum upgrade would it be possible to install the Installer version instead of the Portable one and have my btc transferred from one version to the other ? If yes , how can I manage this?
Simply install the installer version, then copy your old wallet file to the wallet storage directory. For "installer" version, you can find the wallet files here: http://docs.electrum.org/en/latest/faq.html#where-is-my-wallet-file-located