The accurace of secondary structure prediction depends on the method (obviously), but also on the target protein and on the definition of what you call a 'correct prediction'. It is probably around 70-80% correct on a per-residue basis. Maybe others can give you better numbers.
I must caution against 'comparing the 2ndary structure prediction of orthologs'. First, if they are orthologs, chances are good that the secondary structure will be identical. If there are (subtle) differences, they will most likely no be picked up by the prediction methods. It is important to know that the best prediction servers (Predictprotein, PSI-pred, Jpred) do not use a single sequence for the prediction but rather an alignment of related sequences (including orthologs). This is because the conservation properties contain important information that improve structure prediction. For some methods you can (or even have to) provide an alignment directly, while for other methods you submit a single sequence but the programs will then start to search for homologs (and align them) on their own. As a consequence, you cannot expect to see different predictions for related sequences.
Some explanation on the purpose and pitfalls of secondary structure prediction can be read in the manual of Predict Protein in these paragraphs. Programs/servers that you might want to check include PredictProtein, Psipred and jpred.
Tertiary structure prediction ab initio is not ready for prime time,although it is actively being worked on and makes some progress. The reason for this discrepancy is that the problem is much harder (many more parameters to predict) than a pure 2ndary structure prediction. However, a 3D structure prediction is absolutely feasible if you have a good template with known 3D structure, which is related to your protein of interest. Again, by this method you will never be able to see differences between orthologs. If you really want to see subtle differences you have to go for the real structure.
