Hi,
1- Actually, best time to visit Iran is February to May. And It completely depends on your preferences. If you are OK with cold weather, there are tourist attractions in winter, too. However, south of Iran has tropical climate and is warmer in winter. I suppose your plan is visiting tourist attraction in cities, rather than in nature, so you can find activities to do at nights too.
2- 350-400$ (excluding transportation) is a bit tight but not impossible. Assuming 30$ per day (almost according your budget), you can find cheap accommodation in hostel and motels for 7 or 8 $ per night, you can manage your food up to 8$ per day. Therefore, you have 14$ for visiting tickets. In some cities, such as Tehran and Isfahan it would not be enough at all, but you maybe can compensate it in other cities.
3- Most tourist attractions are open in winter. Your examples are quite open, even more attractive in snow. But you have to keep in mind that heavy snow can disturb some activities such as inter-city or intra-city traffic for one or two days (for each region).
4- Yes, there are plenty of them. You can search Iranian booking websites, in English. But maybe, cheaper hostels would not be available on the net.
5- Actually, it is not good enough. But as I said it depends on your preferences.
6- At 10 or 11 o’clock at night, you can have activities to do in metropolises like Tehran. But in smaller cities Like Qom and Kashan I am not sure.

I have to say, your rough plan needs much modification to be practical and on-budget. For example, I suggest you to visit Tehran on your arrival and after visiting Isfahan, take a train straight to Mashhad (passing central and eastern deserts of Iran) instead of coming back to Tehran. Even you can stay one or two days in Tabas to visit “Halvan desert of Tabas.”
Also, I do not know what kind of tourist attractions you prefer. Your given examples (chair lift, different caves, Filband village) are natural sights out of the cities. So, for some of them, such as caves, I suppose you would have difficulties to go there without a rental private car.
However, if you decide to come to Iran, I can help you for your itinerary. And you are welcomed for a one or two-day tour in Tehran by my car for free, since I live in Tehran.
Good Luck

We went for 3 nights, 2 days. The weather was terrible, freezing cold and you could not see the mountains at all because the cloud was so low but it was one of my best trips ever! You can walk down the road to Cat Ba(?) village yourself. It was just amazing! I would do it again in a heartbeat.

Sounds like a terrible plan to me. Weather is of course variable from year to year, but we were all over the Midi-Pyrenées last year at the end of March, beginning of April (both French and Spanish sides), and it was freezing cold, rainy, windy, sleeting, foggy...you name it - and we weren't hiking but rather renting apartments, staying in hotels, visiting spa towns, etc., so did not really have to contend with the elements much. We did very much appreciate Le Bus à Un Euro for transport.

Find somewhere else to carry out this plan. It's not as though you're not spoiled for choices.

...next week will be warmer again. But be aware that weather can change very rapidly in the Alps. Nobody knows whether the first week of Septembe will be sunny and warm or if there will be snowfall at Index and Lac Blanc.

First time I've heard of someone going to Sanlucar de Barrameda, I even had to look it up, lol!

Arcos de la Frontera is one obvious suggestion, being very close to where you are going. Weather depending, you could also visit the white towns of the Grazalema / Ronda area, such as Setenil de las Bodegas, Zahara de la Sierra, etc. If the weather is nice you could go hiking to Garganta Verde, El Torreon, as well as the twin peaks of Simancon/Reloj above Grazalema.

El Caminito del Rey, though refurbished for mass tourism, is also an interesting option. The area of El Torcal makes for an interesting hike.

You could cycle on one of the via verde - old railways turned into cycling paths. Popular ones are around Olvera and Cabra.

If you're going to Almeria, you can also visit the seaside of Cabo de Gata nature park. And on the way there you can go to some white towns of the Alpujarras, such as Capileira and Trevelez.

In northern part, Baeza and Ubeda are both UNESCO listed top sights, but for me personally Cazorla and the mountainous area around is a bigger draw.

A lot will depend on weather, could be +17C and sunny or +11C and rain. I wouldn't have too high expectations.

We don't know what you are interested in, and neither do you until you do some basic research. I'd advise getting a copy of Lonely Planet guidebook which has some basic itinerary suggestions and lots of information re travel.

Happy to thereafter answer specific questions.

But yes, in December you'd likely get warmer weather in the south of the country.

Title:Top 12 Things To Do In Port Issac, Cornwall

Destinations covered: Port Issac, Cornwall

Theme: Destination guide

Style: Travel/Destination guide

Synopsis: Port Issac is a beautiful British seaside holiday destination. A traditional fishing village in north Cornwall and home to the television series Doc Martin, starring Martin Clunes. It is a truly picturesque village with narrow winding streets lined with whitewashed cottages, that head down to the harbour where you can watch local fishermen landing their daily catch of fish, crab and lobsters.

URL:https://www.goliveyoung.com/top-10-things-port-issac/

..... an up-date on the current weather in Morocco, according to the link below very cold weather prevails on high ground and across northern Morocco. While south of the High Atlas, the souther provinces and the Atlantic coast south of the High Atlas have mild and warm temperatures generally. Agadir looks like a good base at the moment.

https://www.bladi.net/maroc-meteo-aujourdhui,53431.html