I find that around moves 8-10 the centre of the board gets quite cluttered.
Typically (exceptions occur) around moves 8-10 you should have completed your development: you should have moved all or most minor pieces (i.e. ca 4-6 moves), should have moved a few pawns to occupy the center and to make space for developing the bishops (ca. 3-5 moves) and have castled (one move).
Yes, pieces and pawns tend to "clutter" in the center because it is an important part of the board and if you control the center you often have an advantage. This cluttering is not a bad thing and should affect both players equally.
I have a hard time finding good places to put my pieces and improve my position, they get in one another's way, and I often find my opponent threatening my pieces with mere pawns, forcing me into awkward moves I don't want to make.
Because there is so much going on in the center, there are often tactics available, particularly in more open positions. So studying tactics is highly recommended for a beginner (and also advanced) player. Closed positions tend to be quieter and are better if you like slow, maneuvering kind of games.
In either case, if you have completed development, (which you should around move 8-10), it is time for the middlegame and you should come up with a plan on how to proceed and also consider the plans of your opponent and if necessary prevent them. Finding plans is not something easy to learn and requires time and experience. I would recommend to start with analyzing the position and to check:
- Are any pieces (mine or opponent) attacked?
- Are there obvious tactics available?
- Which pieces are well placed (lots of space to move around and attack something) and which not (little space to move and don't attack anything)?
- Is the king safe? Typically with many pieces on the board you want it behind pawns on either side after castling.
- Are there any weaknesses in the position such as isolated pawns?
- Who owns the center?
- Are there open or half-open files which are occupied or can be occupied by rooks?
- Is the position open or rather closed? Who would benefit from changing it?
Check all of these items for both sides (you and your opponent). Based on the outcome of this evaluation you should come up with a plan. I ordered the questions in order of importance (exceptions occur), e.g. usually you want to deal with attacked pieces (if you lose/win material) first, before thinking of anything else.
I realise that part of this comes down to choice of opening, and perhaps I haven't read enough and played enough to get a better sense for which openings/defences lead to more open or closed games
is a cluttered board, or a feeling of being cramped also a sign I am trying to develop too many pieces too quickly, or that I am being too hesitant to exchange pawns and pieces in the opening, or even just of a lack of imagination on my part, or anything else along these lines?
I don't think your problem is really related to open/closed positions but rather to a lack of experience of what is possible and knowledge of tactical motives.
For a beginner I would not recommend to worry to much about concrete openings and to study these. Instead I would play openings from general opening principles (occupy the center, develop pieces, ...) and focus on tactics and learning other themes like good bishop/bad bishop, open files (and what to do with them), undermining a closed center, etc.