I think the passage is clearer when the prior two verses are included:
1 Corinthians 2:13–16 (HCSB)
13 We also speak these things, not in words taught by human wisdom,
but in those taught by the Spirit, explaining spiritual things to
14 But the unbeliever does not welcome what comes from God’s Spirit,
because it is foolishness to him; he is not able to understand it
since it is evaluated spiritually.
15 The spiritual person, however, can evaluate everything, yet he
himself cannot be evaluated by anyone.
16 For who has known the Lord’s mind, that he may instruct Him? But we
have the mind of Christ.
Verse 14 explains that those who are unspiritual have no discernment and cannot understand any spiritual matter, and thus are equipped to evaluate neither spiritual things or spiritual people.
Verse 13 explains, on the other hand, that spiritual people are taught by the Spirit to discern spiritual things. In Galatians Paul contrasts desires of the flesh with desires of the Spirit (5:16-17). Thus, the ability to discern things of the Spirit necessarily requires one to discern them against things of the flesh. Hence, I believe, the phrase "all things" is appropriate in verse 15.
Regarding your second question, I do not think any of the above is inconsistent. The preceding verses refer to "spiritual things" and not "all spiritual things". Those spiritual things that relate directly to us and our personal salvation are those that we are able to discern. That does not mean to say, however, that we somehow possess the knowledge that God possesses. John Chrysostom wrote (in the 4th century):
We know the things which are in the mind of Christ, which he has
willed and revealed to us. This does not mean that we know everything
which Christ knows but rather that everything which we know comes from
him and is spiritual.
Homily VII on the Epistles of Paul to the Corinthians
This is also reinforced elsewhere in Scripture (e.g. Romans 11:34; Isaiah 40:13).