You mentioned the PHP version, and while that shouldn’t be an issue, you really need to update that. It’s beyond EOL (end of life). While WordPress will run on it (technically), it does not receive security updates.
There are two things I would look at with your login issue. First, if you have set the plugin for moderated registration. All accounts need to be activated. When you enable the setting it will automatically activate the account that you are logged in as so that you don’t lock yourself out. But if you have more than one admin account, those other accounts are not automatically activated – the plugin does not assume that you want them active or not – so any other admin accounts must be set as active. So in your particular issue, if you enabled this setting and were using a different account to log in, that would have to be activated.
The other possibility would be that you have some type of security plugin that is not compatible. That’s not very common, but it can happen. There are plugins that add captchas and other additional authentication fields to the login. WP-Members does use the wp_login action that most of these use, but it is possible that there’s something out there that isn’t compatible. I would recommend testing without other plugins activated to see if the same problem persists and then reactivate plugins one-by-one to see if you can isolate the problem.