Both you questions can be answered with a clear yes: it is fine both from a grammatical and a stylistic perspective to use the subordinate clause and also the mood is the correct one. Concerning the second question, you could indeed use subjunctive II mood. However, the suggestion in itself is already only a suggestion, so there is no need to further mark it as a potentiality by using subjunctive II. In fact, using a subjunctive typically sounds much more like an overkill and indicative is typically preferred.
Of course, you can add a different modal verb, however, that changes the nuance of the sentence. The two following examples are similar yet distinct:
Ich schlage vor, dass du es nochmal versuchst.
Ich schlage vor, dass du es nochmal versuchen solltest.
Concerning the usage of infinitive structure versus subordinate clause, it depends on whether the recipient of the suggestion is the subject of the subordinate clause or not. For example:
Ich werde der Versammlung vorschlagen, den Antrag abzulehnen.
Ich werde der Versammlung vorschlagen, dass sie den Antrag ablehnt.
Here, the second doesn’t sound as good as the first. Both are possible, though, since the subject of the subordinate clause is the Versammlung, i.e. the recipient of the suggestion. In the following example, however, it’s not possible to use an infinitive construction due to the shift of subject:
Ich werde der Versammlung vorschlagen, dass du aufgenommen wirst.
As long as the subordinate clause stays in passive voice, there is no way to turn it into an infinitive.