I’m afraid I don’t know about Russian courtships. I know a bit about 19th century British courtships and they could be highly structured but overall were rather less so than in some European countries.
The higher the status, of both wealth and title, the more protections were extended to marriageable females to ensure financial and birth status in their future partners. Their meetings with potential marriage partners were limited by the functions they could meet at – balls, dinners, dances, events, etc. – so that the chances of meeting ‘unsuitable’ young men were very restricted. Parents could thus control the sort of people they met and thus were likely to get to know enough to marry. By the 19th century the sort of totally arranged marriages where the young woman was introduced to her potential spouse and was allowed no other choice was rare in Britain even in the highest circles and even where there was intense pressure on two people to marry at least some illusion of choice was allowed. There was no ‘7 visits’ rule as far as I can see, but parental pressure could be exerted to try to force the man to commit to marriage if he expressed particular interest in a woman. For example, the Duke of Westminster became friendly with Constance Cornwallis West and her parents enlisted the Prince of Wales to pressurise him into proposing (as the Cornwallis Wests were not sufficiently wealthy or influential – though of good birth - to be able to do this themselves). Whether this made it happen or not, he eventually did propose and was accepted. There were no set limits on engagements, and these tended to last as long as it took to sort out financial arrangements, purchase a trousseau, and prepare the wedding festivities – several months usually for the rich and titled classes. As the Duke was hugely wealthy and the Cornwallis Wests not well off (at least comparatively) her dowry was pretty nominal, and he would have been expected to contribute to settlements entitling her to support in widowhood, financial arrangements for children etc. This sort of thing happened the other way around where English lords married for money, for example where Consuelo Vanderbilt married the Duke of Marlborough – her dowry was huge and her parents had to settle money on him. In other cases there was a more equitable parental handover of money but very seldom were there no financial arrangement for upper/upper middle class marriages.
There was a greater variation in courtships in the middle classes and the (heavily fictionalised) film about Beatrix Potter gives some indication of this – she and her fiancé Norman Warner met much later than was usual for those of the upper classes, they were allowed to meet (albeit heavily chaperoned) for business purposes, and when they decided to marry there was little her family could do to prevent it apart from exerting the moral pressure to force a long engagement, during which her fiancé died. Otherwise middle class courtships of the 19th century were pretty much like those of the nobility, albeit less structured and formal as you passed down the social scale. Financial status was terribly important in any marriage considerations because of poor contraceptive knowledge and no social security. Therefore families tended to try to pressure their children to stick to their own social class or marry upwards, to ensure maximum financial security. Class was also important – the middle classes may have been fluid but there was a significant difference between a bank clerk and a shop assistant for example, and as Beatrix Potter’s case showed, the Potters considered themselves very much above the Warners in terms of social scale, although their origins were similarly in trade. Engagements tended to be longer in the middle classes (sometimes years) because the man needed to demonstrate he could support a wife and children (the woman might – for example – work by serving in a shop or by the end of the century, work in a variety of occupations ranging from typist to journalist, but she would not do so once she married). Formal financial settlements tended to be limited to the upper middle classes. Again, while there might be huge family pressures for individuals to marry, formal arranged marriages as such were very rare in Britain and the illusion at least of free choice was generally maintained.
There’s actually a huge amount of information about British courtship, engagement and dowry customs in the 19th century, but none of it is in a neat little package. You have to read diaries, biographies, novels and magazines to get an idea of what was going on, and of course it changed as the century progressed. The phenomenon of ‘dollar princesses’ at the end of the century were unknown at the beginning, for example. Then there were the exceptions – the families that didn’t allow the children to be married at all (Elizabeth Barrett Browning’s father springs to mind) for no good reason except control issues. It’s an enormous subject – and that’s just Britain, never mind about Russia!