The short answer to your main question is yes. But we need to understand that God is faithful and true to all who acknowledge Christ and have placed their faith in Him.
2 Timothy 2:11-13 (NIV): “If we died with him, we will also live with him; if we endure, we will also reign with him. If we disown him, he will also disown us; if we are faithless, he will remain faithful, for he cannot disown himself.”
Paul is writing to Timothy to encourage him (and fellow Christians in the Ephesian church) to be faithful. Paul’s second letter to Timothy was written after Paul’s earlier imprisonment in Rome (circa A.D. 62-63) and during his imprisonment under Emperor Nero (circa A.D. 66-67). Paul describes how he was languishing in a cold dungeon, chained up like a common criminal.
This addresses your last question – Paul is not speaking to unbelievers. Paul is speaking to believers who are facing severe persecution and who may, under pressure, disown Christ, like Peter who thrice denied that he knew Jesus. Yet Jesus forgave and restored Peter.
Jesus himself said
“Whoever acknowledges me before men, I will also acknowledge him before my Father in heaven. But whoever disowns me before men, I will disown him before my Father in heaven” (Matthew 10:32-33).
The warning is clear – if we deny Christ or disown him before men, Christ will do likewise with us.
With regard to the last part of the quote, “if we are faithless, he will remain faithful”, the following verse may shed some light:
Romans 3:3-4: “What if some did not have faith? Will their lack of faith nullify God’s faithfulness? Not at all! Let God be true, and everyman a liar.”
God is faithful and true and will keep his promises, and he has promised to
“keep you strong to the end so that you will be blameless on the day of our Lord Jesus Christ. God, who has called you into fellowship with his Son Jesus Christ our Lord, is faithful” (1 Corinthians 1:8-9).
There is therefore every assurance to the believer that God will keep that promise.
Edit: With regard to Revelation 21:8 the context is set in the future with the Holy City, the new Jerusalem, established on earth. The Alpha and the Omega, the Beginning and the End, declares that all who overcome (see the seven lettters in Revelation 2:7, 11, 17, 26; 3:5, 12, 21) will inherit and drink without cost from the spring of the water of life. But the second death awaits the cowardly, the unbelieving, the vile, the murderers, the sexually immoral, those who practise magic arts, the idolaters and all liars.
Revelation 21:8 is not about “the bride of the Lamb” – God’s people – but is a warning for all who are the enemies of God, who have rejected Christ Jesus and who have aligned themselves against Him. They will be cast into the fiery lake of burning sulphur – the second death.
My NIV Study Bible notes say this about Revelation 19:20 which describes how the beast and the false prophet are thrown alive into the fiery lake of burning sulphur:
Punishment by fire is prominent in both biblical and non-biblical Jewish writings (e.g., 1 Enoch 54:1). Originally the site of a cultic shrine where human sacrifices were offered, it came to be equated with the “hell” of final judgment in apocalyptic literature.”
Believers in Christ have nothing to fear.