For your question about strike-throughs, the magic word is "retained".

Retained wires and bristles and other wire-like-things are not exactly common but certainly obtainable. Retained identifiable other things are much rarer.

Actually, there are a couple of other good terms. One is "bonded" and one is "mated"

The "retained" term is usually used when you have something smaller than coin-sized struck into a coin.

The "bonded" term is usually used when you have two items of similar size struck together and there's no point arguing about which is struck into which.

The "mated" term is used for two pieces that were struck together and didn't stick, but are still both available. There are some jaw-droppers in this category, for example:

1973-S $1 Eisenhower Dollar--Indented by 1.73 gm Planchet, Mated Pair--PR67 and PR67 RB NGC. When the planchet for this particular proof Eisenhower dollar entered the press, a smallcopper-based planchet also entered the striking zone between the larger flan and the obverse die. The two resulting pieces, both the indented Eisenhower dollar and the uniface copper-based planchet,are offered here as a mated pair.

The 1973-S Eisenhower Dollar, PR67 offers gleaming fields and excellent preservation in general with minimal patina. A circular indent (from the piece listed below) obscures the vast majority ofEisenhower's head. Interestingly, the reverse is essentially unaffected.

The Uniface Foreign Planchet Indent, PR67 RB, 1.73 gm, shows vivid blue-green, rose-violet, and magenta patina on the struck side, while the reverse, which displays a ghostly echo ofEisenhower's ear, is predominantly copper-orange with small swirls of blue-green and violet. The low weight of the planchet does not correspond to an American issue; the foreign planchet mighthave been intended for a Latin American coinage, such as the El Salvador centavo or the Panama centesimo.

To read the complete lot description, see:
1973-S $1 Eisenhower Dollar--Indented by 1.73 gmPlanchet, Mated Pair--PR67 and PR67 RB NGC.... (https://coins.ha.com/itm/errors/1973-s-1-eisenhower-dollar-indented-by-173-gm-planchet-mated-pair-pr67-and-pr67-rb-ngc/a/454-3523.s)

For a couple others, see:
Multiple Coin Lot...(https://coins.ha.com/itm/errors/mated-pair-1963-50c-half-dollar-double-struck-indented-by-a-cent-planchet-ms62-pcgs-and-a-1963-half-dollar-struck-on-a/a/267-8658.s)
2015 $50 One-Ounce Gold Eagle -- ObverseIndented by Retained Plastic Fragment (https://coins.ha.com/itm/errors/2015-50-one-ounce-gold-eagle-obverse-indented-by-retained-plastic-fragment-first-strike-ms69-pcgs/a/1311-5070.s)

To read the earlier E-Sylum article, see:
VOCABULARY TERM: IMPRESSED ERROR (https://www.coinbooks.org/v23/esylum_v23n09a13.html)