I have noticed a padlock appearing near some URL's. I am also having trouble opening some sites. This related?

[randy4sam]

[DunbarPappy®ϟϟ]

Not necessarily.
Padlocks indicate the site is using SSL which is communication that's encrypted, and it's 'blended' with that sites 'Certificate'. This assures authentication.

This formerly was a common practice on high-value sites, like banking, credit card and the purchase phase of a retail website. Lately, because of hacking proliferation, sites are moving all portions of their website to SSL to assure visitors they are receiving pages just as they are on the server, and have not been altered during transit across the Internet.

"Trouble" opening some sites could be any number of reasons.
If you could be more specific about what is, or is not happening, a more refined answer could be attempted.
(What browser & operating system would also help.)