It is difficult to answer this question, as technical terms may differ between the madhabs and there is no consensus among many terms.
But basically there are five basic terms defining any ruling in fiqh (my own translation take it with care)
I've added links to the translation of the terms, please not that for al-masnoon the best given translation is "Supererogatory", which is not among the first translations mentioned.
قال الشيخ عبد الرحمن السعدي في منهج السالكين:
[الأحكام الخمسة: اَلْوَاجِبُ: وَهُوَ مَا أُثِيبَ فَاعِلُهُ، وَعُوقِبَ تَارِكُهُ وَالْحَرَامُ: ضِدَّهُ. وَالمَكْرُوهُ: مَا أُثِيبَ تَارِكُهُ، وَلَمْ يُعَاقَبْ فَاعِلُهُ. وَالْمَسْنُونُ: ضِدَّهُ. وَالْمُبَاحُ: وَهُوَ اَلَّذِي فِعْلُهُ وتَرْكُهُ عَلَى حَدٍّ سَوَاءٍ. وَيَجِبُ عَلَى اَلْمُكَلَّفِ أَنْ يَتَعَلَّمَ مِنْه ُكُلَّ مَا يَحْتَاجُ
إِلَيْهِ فِي عِبَادَاتِهِ وَمُعَامَلَاتِهِ وَغَيْرِهَ
Sheikh abdurrahman as-Sa'di said in his manhaj as-Salikeen: The five rulings:
- wajib (what is necessary) this includes anything on which the person who performed it would get rewarded and a person who left it would be punished.
- And al-Haram is its opposite (of wajib).
- al-makrooh: includes anything a person who kept away from it or left it would be rewarded and won't be punished if performed it.
- al-masnoon: is the opposite of al-makrooh.
- al-mobaah: is anything where it is indifferent whether one does or left it. And it is necessary for any mukallaf (anybody who is asked -because of age, mind etc.- to follow the orders of shari'a) to learn all from it which he may need in his worships and transactions etc.
based on this fuqaha' of the madhab made some differences:
For example you may find some of them differing between what is called fard/farz and what is called wajib!
Fard has mostly a higher value as it means that it is a clear order! But to be honest I still miss the clear meaning of wajib according this view if fard is set higher!
Also some differ between what is masnoon or sunnah: saying it is sunnah muakkadah or mandub or marghoob (raghibah).
Or they may add adjectives or combination of two (close) rules: like makrooh tahrimi to show that something even if it is only makrooh it has a tendency of being close to haram!
A counterpart for makrooh tahrimi
According to this logic: mandub, sunnah muakadah or raghibah may be an analogue to makrooh tahrimi for some madhabs while the analogue would (I'm not sure as stated above) be wajib for others!
Note that maliki use both terms raghibah (AFAIK only for the fajr sunnah) and sunnah muakadah and i have no idea which one according them is set higher.
Finally what came into my mind right now might be a good counterpart of makrooh tahrimi is the example of fard kifaya: For example the funeral prayer for a Muslim is AFAIK considered as fard kifaya by all madhabs: this means if any (even one) Muslim performed this prayer he would be rewarded but if nobody performed it all Muslims would be punished or considered as having committed a (major) sin! On the other hand we have fard 'ayn ('ayni) for which the Friday prayer stands as an example it is a fard prayer for any Muslim male who is considered as mukalaf and has no valid excuse, so if he doesn't perform it and has no valid excuse he should expect punishment or should consider himself as a sinner.
Your last question: Are there any non-haram acts that carry legal punishment in dunya?
Can be answered based on the view of two madhabs namely the maliki and hanbali with their source of fiqh called sad a-dharaai'i which may allow punishment (or at least a prohibition) for a (usually) halal act, if in a special case this would lead to haram.
I'm quite sure not to have covered all the terms and questions well enough, but if possible I'll improve my answer.