Origi joins Forest's last-minute signings, along with Irish centre-back Andrew Omobamidele, Ivorian midfielder Ibrahim Sangare, England winger Callum Hudson-Odoi, Greek goalkeeper Odysseas Vlachodimos, Argentine defender Nicolas Dominguez and Portuguese left-back Nuno Tavares.
Financial details were not disclosed, but Argentine media reported the loan had a purchase option worth 11 million euros ($11.95 million).
Johnson alleged Podence, who scored Wolves' equaliser, had spat at him in the 90th minute of the match. The incident was checked by VAR at the time and no action was taken against the 27-year-old but the Portuguese player was charged on Wednesday.
Wood, who joined Forest on loan from Newcastle United in January, picked up a thigh injury in a 3-1 loss at Tottenham Hotspur on March 11.
The Ghana captain, a free agent, arrives from Qatari side Al Sadd, where he played since 2021, scoring 15 goals in 21 league matches and winning the Qatar Stars League title last season.
England forward Rashford's superb solo effort after six minutes, dancing past defenders before firing home, took him to 18 goals in all competitions this season. Only Manchester City's Erling Haaland has more for a Premier League club in 2022-23.
Forest will host the first leg after beating Wolverhampton Wanderers in a penalty shootout while Marcus Rashford's late double led Manchester United to a 3-0 victory over third-tier Charlton Athletic on Tuesday to reach the last four.
There were 10 goals and a red card as Aston Villa drew 5-5 at home to Nottingham Forest on Wednesday in England's second tier Championship.
After the fifth round meeting with Championship side (second tier) Hull, Chelsea host Barcelona in the Champions League round of 16 next week.
Karanka, a former Real Madrid and Athletic Bilbao player, was fired by Middlesbrough last season as the club headed towards relegation
FA Cup holders Arsenal suffered their first third round defeat for 22 years as they crashed to 4-2 defeat by second-tier Nottingham Forest