You've asked some great questions and the Holy Spirit is dealing with my heart about some of these same things, so I'll try to focus on what I've been shown (primarily about my own heart) in the hope that it's somewhat beneficial to you.
I don't post often these days but believe this is something very critical, particularly in the West.
So, how does/can a believer know when he is living (for the Lord) in his own strength and when he is living in God's strength?
When I live in your own strength, there's no rest. The focus is oftentimes on performance, even about things the Lord has ordered (i.e. times in the prayer closet, reading the word, witnessing, whatever the Holy Spirit has asked me to do).
There's no pride in simply being a servant and if I have done what the Holy Spirit has ordered, then there's rest, even in the midst of what may appear to be impossible circumstances.
In the modern church, it's quite easy to do it ourselves and then say something like "Thank you Jesus!" when in reality, He didn't deliver us, we delivered ourselves.
If I do what Jesus tells me to do (even if it's somewhat painful to do), let's say confess a sin to a brother or sister or make restitution, then I've found experientially that great joy has risen in my heart. It's quite literally been explosive at times, what a wonderful gift from Jesus...
On the other hand if I do it under my own power, there's no joy but either a fixation on what I've done or an intense desire to move onto the next thing...
What are symptoms that you are walking in the power of your own flesh?
Again, it's primarily a fixation on performance. This can lead to pride, simply because we've been obedient. If I feel pride rising up when I've done something ordered by the Holy Spirit, then that's something I have to take to Jesus. He will purge it if I confess the sin and ask Jesus to take it.
What do you do when you reach an impossible circumstance? Do you prayerfully and joyfully wait on Jesus no matter what happens to deliver you or do you take matters into your own hands? That's a very good barometer of where you're at with Jesus.
And what about when I believe I was led by the Spirit to do something or to a certain realization, and then I start doubting that I might have achieved it all in my own strength?
If there's an absence of joy and a fixation on your performance, it's likely to have been done in your flesh. That's an overly broad generalization but it's something you should take to the prayer closet and lay it out before Jesus. The devil is also a liar as I'm sure you know and he'll often lie. He lies to me all of the time, unless I've chosen not to obey what the Holy Spirit has laid on my heart to do.
If I've done what has been ordered by the Spirit, then I just do it and leave the results to God. I'm accountable to Jesus first and foremost...
How do I fight those doubts, and find freedom and rest in the truth?
Prayer, prayer and more prayer. You're going to need to stand on the promises of God and I would ask you to pray and ask the Holy Spirit to quicken His promises to your heart and then stand on those. As you begin to press the Lord, you're very likely to see some things rise from your heart that you were unaware of, and you're going to have to pray through these. If it's sin, confess it and repent it and throw it aside and move on. If it's a lie, bring that to Jesus as well.
I would encourage you to read and pray over the story of Saul. These "tricks of the flesh" as one of my pastors has phrased it, is readily discernible in the life of Saul.
I pray the Holy Spirit give you discernment when you simply do what He has asked you to do or not to do, which is very important as well. Blessings to you on this pilgrimage...