#1 and #3 are correct. "to recommend (that)" and "to suggest (that)" require the subjunctive. More information here: https://learnersdictionary.com/qa/the-subjunctive excerpt Verbs that often (but not always) are followed by a verb in the subjunctive include: advise, insist, propose, require, ask, intend, recommend, suggest, demand, order, regret, urge, desire, prefer, request, and wish.

1-3. The grammar structure is: subject+RECOMMEND/SUGGEST+ (that)+subject, indirect object+ (should)+ Verb. Althou, is a third person, it doen't use the S. I recommend (that) he (should) continue practicing the four skills. To sum up, should and that is optional.

2 and 4 are correct

Hey, 2 and 4 are both correct but if you could only chose one then I'd go for sentence 4.