Background
First, let be very clear about the operative word ("blessed") here in 1 Peter 1:3 is εὐλογητός - an adjective literally meaning "well spoken of". [It is very different from the other word translated "blessed" used in places like Matt 5 which is the adjective μακάριος = "happy".]
The word εὐλογητός (eulogétos) only occurs 8 times (Mark 14:61, Luke 1:68, Rom 1:25, 9:5, 2 Cor 1:3, 11:31, Eph 1:3, 1 Peter 1:3) and is used EXCLUSIVELY in the NT to refer to The Father and Jesus. According to BDAG it can be translated as either:
blessed, praised ... Of Christ MPol 14:1 ὁ εὐλογητός as a periphrasis for the name of God which is not mentioned out of reverence Mark 14:61.
In the LXX it is also applied to God such as in 1 Kings 1:48, 2 Chron 2:11, 6:4, Ps 71:18.
[The Latin equivalent is used in the Latin Grace before a mean, "Benedictus benedicat." = May the Blessed One [Jesus Christ] bless.]
Answer to OP's Questions:
- In 1 Peter 1:3, the One blessed or praised is "the God and Father of our Lord Jesus Christ!
- The difference between "blessed" and "praise" is documented above - both are as good as English can give for the translation of the Greek εὐλογητός (eulogétos). The Amplified Bible gives this: "Blessed [gratefully praised and adored]" The GNT is one other exception which has: "Let us give thanks to". This is a bit of a linguistic stretch.
- For the distinction between εὐλογητός (eulogétos = well spoken of) and μακάριος = "happy", see above.
- In Heb 7:7 we have the cognate verb εὐλογέω used very extensively (43 times) in the NT. The verb is used often when "praising" God such as James 3:9, John 12:13, Luke 24:53, 19:38, 13:35, 2:28, 1:42, Mark 11:9, Matt 23:39, 21:9, etc. The comment in Heb 7:7 is discussing a unique situation that might be the subject of another question but does not lay down a general rule as shown here. However, it is for this reason that I personally prefer "praise" in this instance rather than "bless" but both are correct.
- In Ps 1;1 we have the word אֶשֶׁר (esher) = happiness, blessedness, and thus is closer in meaning to the Greek μακάριος = "happy".
Note: In Ps 118:26 we have -
Blessed is he who comes in the name of the LORD. From the house of the LORD we bless you.
The highlighted words here are both בָרַךְ (barak) = "to kneel, bless", quite different from that in Ps 1:1 above. It is translated by εὐλογητός (eulogétos) in the NT in Matthew 21:42, Mark 12:10, Luke 20:17, Acts 4:11, and 1 Peter 2:7, where in all these instances, people bless (give honor and glory to) God.